Kick stability in groups and dynamical systems

Leonid Polterovich ^{∗} and Zeev Rudnick School of Mathematical Sciences Tel Aviv University 69978 Tel Aviv, Israel polterov@math.tau.ac.il rudnick@math.tau.ac.il

June 20, 2001

Abstract

We consider a general construction of “kicked systems” which extend the framework of classical dynamics. Let G be a group of measure preserving transformations of a probability space. Given a one-parameter/cyclic subgroup (the flow), and any sequence of elements (the kicks) we define the kicked dynamics on the space by alternately flowing with given period, then applying a kick. Our main finding is the following stability phenomenon: the kicked system often inherits recurrence properties of the original flow. We present three main examples.

1) G is the torus. We show that for generic linear flows, and any sequence of kicks, the trajectories of the kicked system are uniformly distributed for almost all periods.

2) G is a discrete subgroup of P SL(2, ℝ) acting on the unit tangent bundle of a Riemann surface. The flow is generated by a single element of G, and we take any bounded sequence of elements of G as our kicks. We prove that the kicked system is mixing for all sufficiently large periods if and only if the generator is of infinite order and is not conjugate to its inverse in G.

3) G is the group of Hamiltonian diffeomorphisms of a closed symplectic manifold. We assume that the flow is rapidly growing in the sense of Hofer’s norm, and the kicks are bounded. We prove that for a positive proportion of the periods the kicked system inherits a kind of energy conservation law and is thus super-recurrent.

We use tools of geometric group theory (quasi-morphisms) and symplectic topology (Hofer’s geometry).

Key words: sequence of transformations, kicked system, stability, time-reversing symmetry, uniform distribution, lattice, quasi-morphism, Hamiltonian diffeomorphism, Hofer’s metric.

MSC2000: Primary 37Axx, Secondary 11K06, 20F69, 53Dxx, 70Kxx

∗Supported by THE ISRAEL SCIENCE FOUNDATION founded by the Israel Academy of Sciences and Humanities.

1 Introduction and main results

In the present paper we address the following question: ”How far can a flow be kicked?”. More precisely, consider the behavior of a one parameter/cyclic subgroup of a Lie group under the influence of a sequence of kicks. The kicks arrive periodically in time. The kicks are deterministic, while the period is chosen ”at random”. We are interested in the following stability type question: does the kicked system inherit some recurrence properties of the original one? It turns out that in some situations (linear flows on tori, isometries of P SL(2, ℝ)/Γ , ”rapidly growing” Hamiltonian flows on symplectic manifolds) such a stability indeed takes place with positive probability even when the kicks are quite large.

1.1 Sequential systems

Let G be a group. Consider the set G^{∞} = G×G×... of all infinite sequences f_{∗} = {f_{i}}, i ∈ ℕ. Given an action of G on a set X, one can view f_{∗} as a dynamical system (see [BB]): The trajectory of a point x ∈ X is defined as {x_{i}}, where x_{0} = x, x_{1} = f_{1}x, x_{2} = f_{2}f_{1}x and so on. We write f^{(i)} for the evolution of f_{∗} given by f_{i}f_{i−1}...f_{1} and set f^{(0)} = 1l, so x_{i} = f^{(i)}x. Note that the constant sequence f_{∗} = {f} generates iterations of a single map f.

Sequences of maps provide a natural framework for the study of perturbations of the usual settings of dynamical systems ^{1}. We start with the following simple model. Write T for either ℝ or ℤ. Let (h^{t}), t ∈ T be a one-parameter or cyclic subgroup of G which represents a dynamical system on X. Assume that the system is influenced by a sequence of kicks {ϕ_{i}} ∈ G^{∞}. The kicks arrive periodically in time with some positive period τ ∈ T . The kicked dynamics is described by a sequential system f_{∗}^{τ} = {ϕ_{i}h^{τ}}. An orbit {x_{i}} of the kicked system looks as follows: In order to get x_{i} from x_{i−1}, go with the flow for the time τ and then apply the i-th kick: x_{i} = ϕ_{i}h^{τ}x_{i−1}.

There is no reason to expect that general sequential systems have interesting dynamical properties. For instance if G acts by measure preserving transformations, a sequential system may violate the Poincare recurrence theorem etc. However, the kicked systems described above are very special.

Informal Definition 1.1.A. A dynamical property of a subgroup (h^{t}) is called kick stable, if for every sequence of kicks {ϕ_{i}} (possibly satisfying some mild assumptions) the kicked system f_{∗}^{τ} inherits this property for a large set of periods τ. This set of periods may depend on the kicks.

In the present paper we discuss various examples and counter-examples to kick stability. Before presenting a formal definition (see 1.3.A below) let us specify some dynamical properties we wish to deal with and consider a number of examples. The definitions below (possibly with exception of 1.1.C) are straightforward extensions of standard dynamical notions to the framework of sequential systems. Assume that G is a topological group which acts by measure preserving homeomorphisms on a topological space X with a Borel probability measure μ. In 1.1.B - 1.1.D below X is assumed to be compact. Let f_{∗} ∈ G^{∞} be a sequential system.

1In particular, they are used in various models of random dynamics [Ki], see 1.8 below for further comments.

1.1.B. (see [P3]) A sequential system f_{∗} is called strictly ergodic if for every continuous function F on X the Birkhoff sums

\frac{1}{N} ∑_{i=0}^{N−1} F ∘ f^{(i)}

converge uniformly to ∫_{X} F (x)dμ(x).

1.1.C. A continuous function F is called a quasi-integral of f_{∗} if

lim sup_{N→∞} max_{X} \frac{1}{N} ∑_{i=0}^{N−1} F ∘ f^{(i)} > ∫_{X} F (x)dμ(x).

Clearly these definitions complement each other, that is either f_{∗} is strictly ergodic, or it admits a quasi-integral. Sometime it is useful to relax the continuity assumption and to work with characteristic functions χ_{A} of measurable subsets A ⊂ X.

1.1.D. If F = χ_{A} satisfies 1.1.B we say that A is ideally recurrent for f_{∗}, and if F = χ_{A} satisfies 1.1.C then A is called super-recurrent for f_{∗}. Informally speaking, ideal recurrence means that trajectories of the system visit A with the frequency μ(A), while super-recurrence means that there exist arbitrarily long finite pieces of trajectories of f_{∗} which visit A with the frequency > μ(A).

1.1.E. A system f_{∗} is called mixing if for any two L^{2}-functions F and G on X the sequence

∫_{X} F (f^{(i)}x)G(x)dμ(x)

converges to

∫_{X} F (x)dμ(x) ∫_{X} G(x)dμ(x).

1.2 Examples of kick stable flows

Example 1.2.A (stable uniform distribution). Suppose that G is the circle S^{1} = ℝ/ℤ which acts on itself by shifts. Consider the flow

h^{t} : S^{1} → S^{1}, h^{t}x = x + t mod 1.

A characteristic feature of this flow is that for irrational values of τ, every trajectory of the sequential system {h^{τ}} is uniformly distributed in S^{1}, which in our language means that every open interval of S^{1} is ideally recurrent. Take an arbitrary sequence of kicks ϕ_{∗} ∈ (S^{1})^{∞}, and consider the kicked system f_{∗}^{τ} = {ϕ_{i}h^{τ}}.

Theorem 1.2.B. There exists a subset P ⊂ (0; +∞) of full Lebesgue measure such that for every τ ∈ P every trajectory of the kicked system f_{∗}^{τ} is uniformly distributed in S^{1}.

In general, the set P depends on the choice of kicks. Theorem 1.2.B is a consequence of the Weyl criterion. We prove it in a more general context of linear flows on tori in §2 below.

Example 1.2.C (Stable super-recurrence.) Let (X, Ω) be a closed symplectic manifold. Denote by μ the canonical probability measure on X. Let G = Ham(X, Ω) be the group of all Hamiltonian diffeomorphisms of (X, Ω). Recall that a Hamiltonian diffeomorphism is the time-one map of a time-dependent Hamiltonian flow. Such a diffeomorphism preserves the symplectic form Ω. A Hamiltonian function H : X × [0; 1] → ℝ is called normalized if ∫_{X} H(x, t)dμ(x) = 0 for all t ∈ [0; 1]. Every Hamiltonian diffeomorphism h ∈ G can be written as h = h^{1}, where h^{t} is the Hamiltonian flow generated by some normalized Hamiltonian H. In this case we say that h is generated by H. Define a function ρ̄ : G → [0; +∞) as follows:

ρ̄(h) = inf ∫_{0}^{1} max_{x∈X} H(x, t) − min_{x∈X} H(x, t)dt,

where the infimum is taken over all normalized Hamiltonians H which generate h. The function ρ̄ is known as Hofer’s norm (see [H],[P2],[P4]). Let us introduce the following important notion. A sequence {ϕ_{i}} ∈ G^{∞} is called bounded if the sequence {ρ̄(ϕ_{i})} is bounded.

We illustrate stable super-recurrence in the simplest case when X = S^{2} is the unit sphere in ℝ^{3} endowed with the induced Euclidean area form. In this case every diffeomorphism which preserves μ and the orientation is Hamiltonian. Every one-parameter subgroup (h^{t}) of G is simply an autonomous Hamiltonian flow generated by a (uniquely defined) normalized time-independent Hamiltonian function H. Clearly the function H is an integral of motion (the energy conservation law!), and in particular each subset

A_{ϵ} = {H > (1 − ϵ) max H}, ϵ ∈ (0; 1)

is invariant under the flow, and thus super-recurrent in the sense of Definition 1.1.D. Take an arbitrary bounded sequence of kicks {ϕ_{i}} ∈ G^{∞}. ^{2} Consider the kicked system f_{∗}^{τ} = {ϕ_{i}h^{τ}}. Fix ϵ ∈ (0; 1).

Theorem 1.2.D. Suppose that the Hamiltonian H is non-constant, and its maximum set {H = max H} contains a simple closed curve which divides the sphere into two discs of equal areas. Then there exists a subset P ⊂ (0; +∞) whose complement has a finite measure and such that for every τ ∈ P

• the Hamiltonian H is a quasi-integral of the kicked system f_{∗}^{τ};

• the set A_{ϵ} is super-recurrent for f_{∗}^{τ}.

For instance, consider the flow (h^{t}) which rotates each point (x, y, z) ∈ S^{2} with the velocity z around the z-axis. In Euclidean coordinates (x, y, z) on ℝ^{3} it is given by

(1.2.E)  h^{t}(x, y, z) = (x cos(2πtz) − y sin(2πtz), x sin(2πtz) + y cos(2πtz), z).

It is generated by the Hamiltonian function H(x, y, z) = −z^{2} + \frac{1}{3} whose maximum set coincides with the equator {z = 0}. Therefore the theorem above is applicable, and in particular each annulus {|z| < ε} is super-recurrent for the kicked system f_{∗}^{τ} for all values of the period τ ∈ P.

2For instance, one can think that all ϕ_{i}’s are conjugate to elements of some compact (in the C^{∞}-topology)

subset of G.

Theorem 1.2.D is proved in 4.3 below. Even in this simple case the proof we have is not elementary - it is based on powerful methods of modern symplectic topology ^{3} including Hofer’s geometry and Lagrangian intersections. Let us emphasize that our assumption that the curve divides the sphere into two parts of equal area cannot be removed. This is due to the fact that only such curves on S^{2} have the stable Lagrangian intersection property. We refer to section 1.6 for various extensions of this theorem to more general symplectic manifolds.

Example 1.2.F. (stable mixing) Let Γ ⊂ P SL(2, ℝ) be a lattice, that is a discrete subgroup such that the Haar measure μ of the quotient space X = P SL(2, ℝ)/Γ is finite. Let G ⊂ P SL(2, ℝ) be a discrete subgroup, and consider the left action of G on X. Let h ∈ G be an element of infinite order. The Howe-Moore theorem [Zi] (see also 1.5.A below) yields that h is a mixing transformation of X. Let ϕ_{∗} = {ϕ_{i}} be an arbitrary sequence from G^{∞} which represents a finite number of conjugacy classes in G. ^{4} Consider the kicked system f_{∗}^{τ} = {ϕ_{i}h^{τ}}, where τ ∈ ℕ. We say that h is stably mixing if for every sequence ϕ_{∗} as above there exists τ_{0} > 0 such that the kicked system f_{∗}^{τ} is mixing for all τ > τ_{0}. The next result gives a complete description of stably mixing elements of G in purely algebraic terms.

Theorem 1.2.G. Let h ∈ G be an element of infinite order. The following conditions are equivalent:

(i) h is stably mixing on X;

(ii) h is not conjugate to its inverse h^{−1} in G.

For instance, it is easy to see that the time one map of the horocycle flow on X which is given by the matrix

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )

is not conjugate to its inverse already in P SL(2, ℝ) and thus is stably mixing. On the other hand in G = P SL(2, ℤ) every symmetric matrix is conjugate to its inverse by the involution

( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ,

and thus is not stably mixing. A complete description of P SL(2, ℤ)-matrices which are conjugate to their inverses is unknown (cf. [BR]). We refer to 1.7 for the fairly general discussion on the effect of time-reversing symmetry on the kick stability. The proof of Theorem 1.2.G is given in 1.5, 1.7 and §3 below. ^{5} 1.3 Kick stability.

3Dima Burago pointed out that in the case when the sequence of kicks {ϕ_{i}} is contained in a compact subset of G, the conclusions of 1.2.D can be checked by soft methods (compare with the previous footnote).

4For instance, if Γ = P SL(2, ℤ) this assumption holds when all ϕ_{i}’s are non-parabolic and their traces are bounded.

5The condition which tells that the sequence of kicks {ϕ_{i}} represents a finite number of conjugacy classes

Now we are ready to give a formal definition of kick stability which provides a unified framework to examples considered above. In what follows T stands either for ℝ or for ℤ, and T_{+} = {t ∈ T | t > 0}. We fix a class B of subsets of T_{+} of ”large measure”. For instance we can assume that when T = ℝ (resp. T = ℤ) the class B consists of all subsets of T_{+} whose complement has finite Lebesgue measure (resp. is a finite subset).

Let G be a group, and let (h^{t}), t ∈ T be a subgroup which is assumed to be either one parameter (T = ℝ) or cyclic (T = ℤ). This subgroup represents the unperturbed dynamical system with continuous or discrete time. Fix a subset P ⊂ G^{∞} which should be thought of as the set of all sequential systems with a given property. We say that our subgroup (h^{t}) has this property if the set

{τ ∈ T_{+} | the sequence {h^{τ}} ∈ P}

has large measure, that is belongs to B.

Let Φ ⊂ G^{∞} be a set of admissible kicks. Take ϕ_{∗} ∈ Φ and write f_{∗}^{τ}(ϕ_{∗}) for the kicked system {ϕ_{i}h^{τ}}. Denote by P (ϕ_{∗}) the set of all positive values of the period τ such that the kicked system has the given property:

P (ϕ_{∗}) = {τ ∈ T_{+} | f_{∗}^{τ}(ϕ_{∗}) ∈ P}.

Formal Definition 1.3.A. A property of subgroup (h^{t}) is kick stable if for every admissible sequence of kicks ϕ_{∗} ∈ Φ the set P (ϕ_{∗}) of ”good” periods belongs to the class B of subsets of large measure.

Let us illustrate this definition. In Example 1.2.A above, all kicks are admissible so the set Φ coincides with (S^{1})^{∞}. The set P consists of all sequences from (S^{1})^{∞} whose orbits are uniformly distributed in S^{1}. Theorem 1.2.B implies that the property all orbits are uniformly distributed in S^{1} is kick stable for (h^{t}). In Example 1.2.C the set Φ of admissible kicks consists of all sequences which are bounded in Hofer’s norm. The set P is formed by all sequential systems for which the set A_{ϵ} is super-recurrent. Theorem 1.2.D states that the property A_{ϵ} is a super-recurrent set is kick stable for (h^{t}). In Example 1.2.F all sequences of kicks which represent a finite number of conjugacy classes are admissible, and the set P consists of all mixing systems. Theorem 1.2.G tells us when mixing is a kick stable property of the cyclic subgroup (h^{t}).

1.4 Sub-additive functions

We do not know of an argument which provides a unified explanation of the kick-stability phenomenon in all the examples presented in section 1.2 above. Interestingly enough, however that our approaches to Theorems 1.2.D and 1.2.G have a common ingredient. Namely, the desired kick stability is closely related to the geometric behaviour of the corresponding subgroups at infinity.

in G is essential as the next simple example shows. Let G = P SL(2, ℤ) and h = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Choose a surjective function α : ℕ → ℕ such that α^{−1}(τ) is an infinite subset for every τ ∈ ℕ. Put q_{k} = kα(k) − (k − 1)α(k − 1), where k ∈ ℕ. Consider a sequence of kicks ϕ_{k} = h^{−q_{k}}. Clearly it represents an infinite number of different conjugacy classes in P SL(2, ℤ). The evolution of the kicked system is given by f^{(k)}(τ) = h^{k(τ−α(k))}. Thus for every τ ∈ ℕ equality f^{(k)}(τ) = 1l holds for an infinite number of k’s, so the kicked system is not mixing.

Definition 1.4.A. Let G be a group. A function ρ : G → [0; +∞) is called sub-additive if there exists a number C ≥ 0 such that the following holds:

• |ρ(hgh^{−1}) − ρ(g)| ≤ C for all g, h ∈ G;

• ρ(gh) ≤ ρ(g) + ρ(h) + C for all g, h ∈ G.

If ρ is sub-additive then, as is well known, for every h ∈ G, the limit

ρ_{∞}(h) = lim_{n→+∞} \frac{1}{n}ρ_{∞}(h^{n})

does exist. In the next two sections we discuss two applications of sub-additive functions on groups to kick stability.

1.5 Stable mixing and quasi-morphisms

In this section we outline the proof of the sufficient condition of stable mixing for an element of the discrete group G given in 1.2.G above.

We start with a following more general situation. Suppose that D is a non-compact simple Lie group with finite center. Let Γ ⊂ D be a lattice, that is a discrete subgroup such that the Haar measure of the quotient space X = D/Γ is finite. The group D acts on X on the left by transformations preserving the Haar measure. The key ingredient of our approach to stable mixing is the Howe-Moore theorem which provides a link between geometry and dynamics of sequential systems in this setting. In order to formulate it we need the following notions. Let {f^{(i)}} be a sequence of elements of D. We say that {f^{(i)}} goes to infinity if for every compact subset Q ⊂ D there exists i_{0} such that f^{(i)} ∉ Q for all i > i_{0}. From general considerations, it follows that a mixing sequence of elements of D necessarily goes to infinity ^{6}. Conversely, we have:

Howe-Moore Theorem 1.5.A. ([Zi]). Let f_{∗} be a sequential system from D^{∞}. If its evolution {f^{(i)}} goes to infinity then f_{∗} is mixing.

Let G ⊂ D is a discrete group. Below we focus on the action of G on X. Consider the cyclic subgroup generated by an element h ∈ G. Let ϕ_{∗} be an arbitrary sequence from G^{∞} whose entries represent a finite number of conjugacy classes in G. Consider the kicked system f_{∗}^{τ} = {ϕ_{i}h^{τ}}, where τ ∈ ℕ. The next result provides a sufficient condition for kick stability of mixing for our subgroup.

6Indeed, otherwise there is a subsequence which is bounded and therefore there is a convergent subsequence, which is also mixing. Thus (after passing to this subsequence) we have a sequence f^{(i)} → f_{∞} and that is still mixing. Now take a real valued function G, not identically zero, with ∫_{X} G(x)dμ(x) = 0 and set F (x) := G(f_{∞}^{−1}x). By the mixing property of the sequence, we have

∫_{X} F (f^{(i)}x)G(x)dμ(x) → ∫_{X} F (x)dμ(x) ∫_{X} G(x)dμ(x) = 0 while because f^{(i)} → f_{∞}, we have

∫_{X} F (f^{(i)}x)G(x)dμ(x) → ∫_{X} F (f_{∞}x)G(x)dμ(x) = ∫_{X} G(x)^{2}dμ(x) > 0 which gives a contradiction.

Theorem 1.5.B. Assume that there exists a sub-additive function ρ on G such that ρ_{∞}(h) > 0. Then there exists τ_{0} > 0 such that the kicked system f_{∗}^{τ} is mixing for all τ > τ_{0}.

Here τ_{0} depends on ϕ_{∗}. It turns out that the geometric assumption ρ_{∞}(h) > 0 guarantees that for large periods τ the evolution of the kicked system goes to infinity , thus the statement follows from 1.5.A. The details of this argument are given in 3.1 below.

There exists a useful class of sub-additive functions which arise naturally in the bounded cohomology theory of discrete groups (see [Br],[BG],[Ba], [Pi]).

Definition 1.5.C. A function r : G → ℝ is called a quasi-morphism if there exists a constant C > 0 such that

|r(gh) − r(g) − r(h)| ≤ C

for all g, h ∈ G.

Given a quasi-morphism r and an element g ∈ G , there exists the limit

r_{∞}(g) = lim_{n→+∞} \frac{r(g^{n})}{n}.

Note that r_{∞} is homogeneous, that is r_{∞}(g^{k}) = kr_{∞}(g) for all k ∈ ℤ. Thus if g has finite order then r_{∞}(g) = 0. Moreover, r_{∞} is a class function: r_{∞}(hgh^{−1}) = r_{∞}(g). It follows immediately from the definition that if r is a quasi-morphism then the function ρ(g) = |r(g)| is sub-additive in the sense of Definition 1.4.A and moreover ρ_{∞}(g) = |r_{∞}(g)| for all g ∈ G.

Theorem 1.5.D. Let G ⊂ P SL(2, ℝ) be a discrete group and h an element of infinite order in G. The following conditions are equivalent:

(i) there exists a quasi-morphism r : G → ℝ such that r_{∞}(h) > 0;

(ii) h is not conjugate to its inverse h^{−1} in G.

Proof of “1.2.G(ii) implies 1.2.G(i)”: The desired statement is an immediate consequence of 1.5.D and 1.5.B. □

Theorem 1.5.D is proved in 3.2 below (see Remark 3.2.F for references and generalizations of this result).

1.6 Stable super-recurrence in Hamiltonian dynamics

Let (X, Ω) be a closed symplectic manifold. Denote by μ the canonical probability measure on X. Let G = Ham(X, Ω) be the group of all Hamiltonian diffeomorphisms of (X, Ω). Define the positive part of Hofer’s norm ρ : G → [0; +∞) as follows:

ρ(h) = inf ∫_{0}^{1} max_{x∈X} H(x, t)dt,

where the infimum is taken over all normalized Hamiltonian functions H : X × [0; 1] → ℝ which generate h (cf. 1.2.C above). It is an easy exercise to check that ρ is sub-additive (here the constant C of 1.4.A is simply 0). Moreover, the following obvious inequality holds: ρ̄(h) ≥ ρ(h) + ρ(h^{−1}) for all h ∈ G. ^{7}

Let (h^{t}) be a one-parameter subgroup of G. It is generated by some uniquely defined time-independent Hamiltonian H : X → ℝ with zero mean. The law of energy conservation yields that H is an integral of motion. Note that ρ(h^{t}) ≤ t max H for all t > 0, and thus we have ρ_{∞}(h^{1}) ≤ max H. Let ϕ_{∗} ∈ G^{∞} be an arbitrary bounded sequence of kicks (see 1.2.C above). Consider the kicked system f_{∗}^{τ} = {ϕ_{i}h^{τ}}.

Theorem 1.6.A. (stable energy conservation law). Assume that ρ_{∞}(h^{1}) > 0. There exists a subset P ⊂ (0; +∞) of density at least

\frac{ρ_{∞}(h^{1})}{max H}

such that for every τ ∈ P the Hamiltonian H is a quasi-integral of the kicked system f_{∗}^{τ}.

This result gives a probabilistic interpretation for a purely geometric quantity ρ_{∞}(h^{1}). Further, this quantity contains interesting information about kick stable super-recurrence of (h^{t})-invariant sets of the form

A_{ϵ} = {H > ϵ max H}.

We present here two sample results.

Theorem 1.6.B. Assume that ρ_{∞}(h^{1}) = max H. Fix ϵ > 0. There exists a subset P ⊂ (0; +∞) of density 1 such that for every τ ∈ P the set A_{ϵ} is super-recurrent for the kicked system f_{∗}^{τ}.

One can check (see 4.2, 4.3 below) that in Example 1.2.C above the assumption ρ_{∞}(h^{1}) = max H is satisfied, thus (a slightly weaker version of) Theorem 1.2.D follows from 1.6.B.

Theorem 1.6.C. Assume that ρ_{∞}(h^{1}) ≥ 0.9 max H, and max H = − min H. Then there exists a subset P ⊂ (0; +∞) of density at least 0.4 such that for every τ ∈ P the set A_{0.4} is super-recurrent for the kicked system f_{∗}^{τ}.

We prove more general versions of these theorems in §4 below.

1.7 An obstruction to kick stability

We describe here a method of constructing kick unstable systems in a number of interesting situations. Let (h^{t}), t ∈ T be a one-parameter/cyclic subgroup of a group G.

Definition 1.7.A. An element θ ∈ G is called a time-reversing symmetry ^{8} for (h^{t}) if θh^{t}θ^{−1} = h^{−t} for all t ∈ T . Let us introduce the following notation. Given a system f_{∗} ∈ G^{∞}

we write f^{∗} = {f^{(i)}} for its evolution f^{(i)} = f_{i}...f_{1}.

7In fact, in all known examples one has the equality!

8See [LR] for a discussion on time-reversing symmetries and their impact on dynamics.

1.7.B. Creating periodic behaviour. Suppose that (h^{t}) admits a time-reversing symmetry θ. Take a sequence of kicks

ϕ_{∗} = {θ^{−1}, θ, θ^{−1}, θ, ...}.

Consider the kicked system f_{∗}(τ) = {ϕ_{i}h^{τ}}. Its evolution f^{∗}(τ) is 2-periodic:

f^{∗}(τ) = {θ^{−1}h^{τ}, 1l, θ^{−1}h^{τ}, 1l, ...}.

Assume that G acts by measure-preserving homeomorphisms on a topological space X. Obviously, for all τ ∈ T every subset A of X with μ(A) < 0.5 is super-recurrent for the kicked system f_{∗}(τ).

Example 1.7.C. (cf. 1.2.A above). Consider the action of the orthogonal group O(2) on the circle S^{1}. The uniform distribution property of the flow h^{t}x = x + t is not kick stable for kicks from (O(2))^{∞}. Indeed, the transformation θ : x → −x is a time-reversing symmetry for h^{t}. Comparing this with Theorem 1.2.B above we see that the flow h^{t} loses stability when one replaces the group S^{1} = SO(2) by a larger group O(2). This stability breaking mechanism can be observed in many other situations (see 1.7.H and 4.6.A below).

1.7.D. Suppose now in addition that every point x ∈ X admits a nested system of open neighborhoods U_{δ}, δ ∈ (0; δ_{0}) such that ∩_{δ}U_{δ} = {x} and μ(U_{δ}) = δ. We claim that for every τ the kicked system f_{∗}(τ) constructed above is not mixing. Indeed, fix arbitrary τ ∈ T , and take δ > 0 small enough. Since θ ≠ h^{τ} there exists an open subset U of X of measure δ such that θ^{−1}h^{τ}U ∩ U = ∅. Let F be a characteristic function of U. The only limit points of the sequence ∫_{X} F (f^{(i)}(τ)x)F (x)dμ(x) are 0 and δ, while ( ∫_{X} F (x)dμ(x))^{2} = δ^{2}. Therefore the kicked system is not mixing.

Proof of “1.2.G(i) implies 1.2.G(ii)”: Consider the action of G on X = P SL(2, ℝ)/Γ. Let h ∈ G be an element of infinite order. If h is conjugate to its inverse (that is it admits a time-reversing symmetry) then the argument 1.7.D above shows that h is not stably mixing. This completes the proof of 1.2.G. □

1.7.E. Creating random behaviour. Exactly as in examples above, a time-reversing symmetry provides an obstruction to stable super-recurrence in Hamiltonian dynamics. We work in the setting of 1.6, assuming in addition that the group of Hamiltonian diffeomorphisms G is C^{∞}-closed in the group of all smooth diffeomorphisms of X. ^{9} The next result is proved in 4.6 below.

Proposition 1.7.F. Assume that (h^{t}) is a one-parameter subgroup of the group G of Hamiltonian diffeomorphisms of X which admits a time-reversing symmetry. Then there exists a bounded sequence ϕ_{∗} = {ϕ_{i}} ∈ G^{∞} such that the kicked system {ϕ_{i}h^{τ}} is strictly ergodic for all τ ∈ (0; +∞).

Example 1.7.G (cf. 1.2.C above.) Let X be the 2-sphere, and let h^{t} be the circle action which rotates the sphere with constant speed around the z-axes. This flow has no stably super-recurrent invariant sets with piece-wise smooth boundary. Indeed, it has a timereversing symmetry (for instance, the reflection around the x-axes).

9This is true for many symplectic manifolds, for instance if H^{1}(X, ℝ) = 0 or if the cohomology class of the symplectic form Ω is rational. The famous Flux Conjecture (see e.g. [LMP]) which states that this is true for all (X, Ω), is still open.

Example 1.7.H (cf. 1.2.C above.) Assume now that h^{t} is the flow given by (1.2.E) above. Instead of the group of all Hamiltonian diffeomorphisms of S^{2} consider the larger group of all measure-preserving diffeomorphisms ^{10}. It turns out that after such an enlarging of the ambient group, the flow (h^{t}) gets a time-reversing symmetry- the reflection over the (x, y)-plane. A little modification of 1.7.F above yields that (h^{t}) loses stable super-recurrence (cf. 1.7.C above).

1.8. Discussion and open problems

1.8.A. Historical and bibliographical remarks.

Sequential systems arise naturally in random dynamics [Ki]. In the deterministic language they form a particular case of a skew-product. The study of ergodic properties of individual sequential systems was initiated by Bergelson and Berend [BB]. Some special classes of sequential systems were known for a long time. For instance, sequences of Möbius transformations were considered in connection with analytic continued fractions as well as with the discrete Schrödinger operator. Another source of examples comes from non-autonomous ODE’s, see [S]. Very recently Zeghib [Ze] investigated sequences of isometries of a Lorentz manifold. He found that the corresponding dynamics is closely related to asymptotic geometry of the isometries group (cf. 1.5-1.6 above). It is clear that sequential dynamics have not been systematically studied yet, and many natural and interesting questions are still unexplored.

Kicked systems were intensively studied by physicists in the classical (= non-sequential) framework. This class of systems includes a number of famous maps which attracted a lot of attention in conservative chaotic dynamics, such as the kicked top [HKS], the kicked harmonic oscillator (or the Henon map [He]) and the kicked rotator (or the standard map [Ha]). Physicists, however considered these maps from a viewpoint which essentially differs from ours. In order to illustrate the difference, let us return to the the flow (h^{t}) given by (1.2.E) above. Take the constant sequence of kicks ϕ_{i} ≡ ϕ, where ϕ(x, y, z) = (−z, y, x). The corresponding kicked system describes iterations of the single map ϕh^{τ}, which is nothing else but the kicked top map (see [HKS]). In contrast to our setting, in the physics literature ϕ describes the top, while h^{τ} stands for the kick! Further, computer experiments performed in [HKS] suggest that for large values of τ the kicked system is chaotic: a generic trajectory of the kicked top map is uniformly distributed on a huge subset of the sphere. On the other hand, Theorem 1.2.D above guarantees super-recurrence for all positive τ outside the set of finite measure. Thus chaos can coexist with super-recurrence. In fact, the deterministic behaviour of a generic trajectory of the unperturbed system is a kick unstable property. This kick instability, reflected in the transition to chaos for large values of τ, is the main attraction for physicists. The main message of our theory is that even for large τ some (non-generic!) trajectories are still super-recurrent. Indeed, our notion of super-recurrence takes into account behaviour of all (vs. almost-all) trajectories of a system. For instance, the original kicked top map is super-recurrent simply due to existence a fixed point (1, 1, 0). It surprising however that this super-recurrence persists when the constant sequence of kicks is replaced by an arbitrary bounded non-constant sequence. Let us mention also that this conflict between ”all” and ”almost all” is a reflection of the striking difference between L^{p}− and C^{0}− measurements in symplectic topology, see for instance [EP],[P4]. A similar analysis can be carried out for the standard map.

10Besides the Hamiltonian ones, this group includes orientation reversing diffeomorphisms preserving the measure.

Kick stability, the central notion promoted in the present paper, lies in a long series of attempts to formalize robustness of dynamical and ergodic properties of flows and maps (such as recent works on stable ergodicity, see [FP] and references therein). Among its cousins one may recall stochastic stability [V] which naively speaking means stability with respect to small random sequential kicks. It would be interesting to compare kick stability (where the kicks are deterministic and not assumed to be small) with stochastic stability in more detail.

1.8.B. Discrete vs. continuous

In basic examples considered above- uniform distribution on S^{1}, super-recurrence in Hamiltonian dynamics and mixing on P SL(2, ℝ)/Γ - one can address both discrete and continuous versions of the question on kick stability. In the present paper we worked out the continuous versions in the first two examples, and the discrete version in the last one. What happens with the remaining cases? It turns out that cyclic subgroups of S^{1} do not have kick stable uniform distribution property. This was noticed by Dima Burago whose argument is presented in 2.3 below. Further, nothing is known to us about kick stable super-recurrence for cyclic subgroups of the group of Hamiltonian diffeomorphisms of a symplectic manifold. It would be interesting to make some progress in this direction. Finally, we arrive at the kick stability question for flows on P SL(2, ℝ)/Γ. It deserves a special discussion.

Consider the action of G = P SL(2, ℝ) on X = P SL(2, ℝ)/Γ, where Γ is a lattice. It follows from 1.5.A that every non-compact one-parameter subgroup of G is mixing. Is this property kick stable? For instance, the geodesic flow

h^{t} = ( \begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix} )

is not stably mixing. Indeed, it is given by a symmetric matrix and thus admits a timereversing symmetry (see discussion in 1.7.D above).

Question 1.8.C. Is the horocycle flow

h^{t} = ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} )

stably mixing?

Note that the horocycle flow does not admit a time reversing symmetry in P SL(2, ℝ). Being unable to find the complete answer, we present some partial results (suggesting the affirmative solution) and more discussion in the Appendix to §3 below. In particular, we present a link between this problem and spectral theory for the discrete Schrödinger equation.

1.8.D. Generalizations to other dynamical systems.

It would be interesting to investigate stable mixing for lattices in semi-simple Lie groups of higher rank. Does there exists a complete description of stably mixing elements (if any) similar to Theorem 1.2.G above? Recently Burger and Monod [BM1, BM2] showed that unlike the rank-one case, there are few quasi-morphisms of lattices in higher rank groups. Thus new ideas are needed in order to understand that case.

Another potential source of kick-stable systems might be provided by hyperbolic theory. Here is a warm up question: under which conditions is the existence of a hyperbolic attractor a kick stable property?

CONTENTS

Section 2. Linear flows on tori.

Section 3. Detecting stable mixing on P SL(2, ℝ)/Γ.

Section 4. Stable super-recurrence in Hamiltonian dynamics. Acknowledgments: We thank Dima Burago for a number of important critical remarks and suggestions, and in particular for explaining us a construction presented in 2.3 below. We are grateful to Anna Dioubina for pointing out an error in the original version of Remark 3.3.E, as well as to Brian Bowditch and Iosif Polterovich for illuminating consultations on reference [EF]. We thank Leonid Pastur and Misha Sodin for various useful discussions.

2 Linear flows on tori

2.1. Generic flows are kick stable.

In this section we consider kick-stability for linear flows on the d-dimensional torus 𝕋^{d} = ℝ^{d}/ℤ^{d}. It turns out that some very classical results on the “metric” theory of uniform distribution immediately imply that the kicked flows are stably uniformly distributed for almost all periods.

Precisely, for ω ∈ ℝ^{d} consider the “Kronecker map” on the torus 𝕋^{d} given by x ↦ x + ω mod 1. It gives a one-parameter group (t ∈ ℝ)

h^{t}(x) = x + tω mod 1 .

Given β_{i} ∈ 𝕋^{d}, we get “kicks” ϕ_{i}(x) = x + β_{i} mod 1. Put f_{i}^{τ} = ϕ_{i}h^{τ}, so that f_{i}^{τ}(x) = x + β_{i} + τω mod 1. The evolution of the kicked system is then given by

f^{(k)} = f_{k}^{τ}f_{k−1}^{τ} … f_{1}^{τ} : x ↦ x + α_{k} + kτω mod 1

where α_{k} = β_{1} + ⋯ + β_{k}.

We will say that ω = (ω_{1}, …, ω_{d}) is a “generic” vector if its components ω_{i} are linearly independent over the rationals.

Theorem 2.1.A. Suppose that ω is a “generic” vector. Then for almost all periods τ, the orbits {f^{(k)}(x)}_{k=1}^{∞} are uniformly distributed in 𝕋^{d}.

This result is a version of a “metric theorem” of Weyl from 1916 (see [KN]). For the sake of completeness we recall the argument. In the following, given two sequences a(N) and b(N) we will use the notation a(N) ≪ b(N) to mean that there is a constant c > 0 so that a(N) ≤ cb(N) for all N sufficiently large.

2.2. Proof of 2.1.A:

Since 𝕋^{d} acts transitively on itself by translations, it suffices to consider the base point 0 instead of arbitrary x. It also suffices to fix a finite interval [a, b] and show that the result holds for almost all τ ∈ [a, b].

Define normalized “Weyl sums”

S_{h}(N, τ) := \frac{1}{N} ∑_{k=1}^{N} e^{2πih·f^{(k)}(0)}

The basic tool is Weyl’s observation that for uniform distribution, it suffices to show that the normalized Weyl sums converge to zero for all integer vectors h ≠ 0.

To do that, one shows (see below) that for fixed h ≠ 0, one has

(2.2.A)  ∫_{a}^{b} |S_{h}(N, τ)|^{2} dτ ≪ \frac{log N}{N} .

Thus for the sequence of squares N = n^{2}, we have

∑_{n=1}^{∞} ∫_{a}^{b} |S_{h}(n^{2}, τ)|^{2} dτ < ∞ .

By Fatou’s lemma, it follows that ∑_{n=1}^{∞} |S_{h}(n^{2}, τ)|^{2} is integrable on [a, b], and so is finite for all τ in a set of full Lebesgue measure P_{h}. Thus the n-th term tends to zero for all τ ∈ P_{h}. Intersecting over all integer vectors h ≠ 0 we get one set P of full measure which works for all h ≠ 0, that is for all τ ∈ P

S_{h}(n^{2}, τ) → 0, n → ∞ .

Now given any N, find n so that n^{2} ≤ N < (n + 1)^{2}. Writing N = n^{2} + k, 0 ≤ k ≤ 2n we have by using the trivial bound |e^{2πix}| ≤ 1 that

|S_{h}(n^{2} + k, τ) − S_{h}(n^{2}, τ)| ≪ \frac{k}{n^{2}} ≪ \frac{1}{n} → 0, n → ∞

for τ ∈ P and since S_{h}(n^{2}, τ) → 0, we get S_{h}(N, τ) → 0 for all N → ∞.

To show 2.2.A, we square out the sum and directly integrate to get

∫_{a}^{b} |S_{h}(N, τ)|^{2} dτ = \frac{1}{N^{2}} ∑_{k,l≤N} e^{2πih·(α_{k}−α_{l})} ∫_{a}^{b} e^{2πi(k−l)h·ωτ}dτ .

The “diagonal” terms k = l give a total contribution of (b − a)/N to the sum, so to prove 2.2.A it suffices to bound the off-diagonal terms k ≠ l.

Since ω · h ≠ 0 for integer h ≠ 0 (that was the assumption on ω), each off-diagonal term contributes

e^{2πih·(α_{k}−α_{l})} \frac{e^{2πi(k−l)h·ωb} − e^{2πi(k−l)h·ωa}}{2πi(k − l)h · ω} .

Taking absolute values and summing over all pairs 1 ≤ k ≠ l ≤ N gives a contribution bounded by a constant times

(2.2.B)  \frac{1}{N^{2}} ∑_{1≤k≠l≤N} \frac{1}{|k − l|}.

For fixed n ≠ 0, the number of solution of k − l = n with 1 ≤ k ≠ l ≤ N is N − |n| if 1 ≤ |n| ≤ N − 1 and zero otherwise. Thus 2.2.B is given by

\frac{2}{N^{2}} ∑_{n=1}^{N−1} \frac{N − n}{n} ≪ \frac{log N}{N}

which proves 2.2.A and the Theorem. □

2.3 Cyclic subgroups of S^{1} are kick unstable.

We present here a counter-example to kick stability constructed by Dima Burago. In what follows we identify Kronecker maps x → x + b with the corresponding elements b ∈ S^{1}. Fix an irrational element ω ∈ S^{1}. The corresponding cyclic subgroup is uniformly distributed in S^{1}. We claim that there exists a sequence of kicks {β_{i}} ∈ (S^{1})^{∞} such that for every τ ∈ ℕ the evolution of the kicked system is not uniformly distributed in S^{1}. Recall from 2.1 that this evolution is given by f^{(k)} = α_{k} + kτω mod 1, where α_{k} = β_{1} + ... + β_{k}. In order to prove the claim, choose a function u : ℕ → ℕ such that the preimage u^{−1}(k) ⊂ ℕ of every integer k ∈ N is a subset of strictly positive density. Put now α_{k} = −u(k)kω mod 1, and β_{k} = α_{k} − α_{k−1}. Fix τ ∈ ℕ and consider the sequence {f^{(k)}}. Since every element of this sequence with k ∈ u^{−1}(τ) vanishes, and the set u^{−1}(τ) has positive density in ℕ, we conclude that this sequence is not uniformly distributed in S^{1}. This completes the proof of the claim.

3. Detecting stable mixing on P SL(2, ℝ)/Γ

In this section we prove Theorems 1.5.B,D and in an Appendix, present some partial answers on Question 1.8.C.

3.1. Proof of 1.5.B:

Let C be the constant from Definition 1.4.A. Put C_{1} = max_{j∈ℕ} ρ(ϕ_{j}^{−1}). The maximum is finite since {ϕ_{j}} represents a finite number of conjugacy classes, and ρ is bi-invariant up to C (see 1.4.A). Denote by f^{(k)}(τ) the evolution of the kicked system,

f^{(k)}(τ) = ϕ_{k}h^{τ} … ϕ_{1}h^{τ} .

Note that for every k > 0 and τ ∈ (0 + ∞)

h^{τk} = f^{(k)}(τ) · ∏_{j=1}^{k} h^{−τj}ϕ_{j}^{−1}h^{τj} .

Applying ρ to both sides of this equation and using properties listed in 1.4.A we get that

(3.1.A)  ρ(h^{τk}) ≤ ρ(f^{(k)}(τ)) + ∑_{j=1}^{k} ρ(ϕ_{j}^{−1}) + 2Ck

≤ ρ(f^{(k)}(τ)) + k(2C + C_{1}) .

Choose τ_{0} > 0 so large that ρ(h^{τk}) ≥ 0.5τkρ_{∞}(h) for all τ > τ_{0} and k ∈ ℕ. Put C_{2} = 0.5τ_{0}ρ_{∞}(h)−(2C +C_{1}). Increasing if necessary τ_{0} we assume that C_{2} > 0. In view of (3.1.A) we have ρ(f^{(k)}(τ)) ≥ C_{2}k for τ > τ_{0}. Thus for τ ≥ τ_{0} the sequence f^{(k)}(τ) goes to infinity (see 1.5). Applying the Howe-Moore theorem 1.5.A we see that for τ > τ_{0} the kicked system is mixing. □

3.2. Quasi-morphisms

Our purpose is to prove Theorem 1.5.D, that is if G ⊂ P SL(2, ℝ) is a discrete group, and g ∈ G an element of infinite order, then the following are equivalent:

(i) There exists a quasi-morphism r : G → ℝ so that r_{∞}(g) ≠ 0.

(ii) g is not conjugate to its inverse g^{−1} (in which case we say that g does not admit a time-reversing symmetry in G).

One direction is immediate: Recall that for a quasi-morphism r, its stabilization r_{∞} is a homogeneous class function on G. Hence if g is conjugate to g^{−1} then r_{∞}(g) = r_{∞}(g^{−1}) = −r_{∞}(g) and so r_{∞}(g) = 0.

Let us show now that (ii) yields (i). Thus if G is a discrete subgroup of P SL(2, ℝ), we wish to show that given any element of G of infinite order which is not conjugate in G to its inverse, there is a homogeneous quasi-morphism r = r_{∞} of G for which r_{∞}(g) ≠ 0.

Remark. Concerning the condition that g is conjugate in G to its inverse g^{−1}, we note that this can only happen for hyperbolic g since elliptic and parabolic elements of P SL(2, ℝ) are never conjugate to their inverses in P SL(2, ℝ). Moreover, if g ∈ P SL(2, ℝ) is hyperbolic then it can be shown that any time reversing symmetry K of g (i.e. an element K such that g = Kg^{−1}K^{−1}) must satisfy K^{2} = −1 (in SL(2, ℝ)). Thus for many cases of interest, such as surface groups, which do not have elliptic elements, this possibility does not arise.

We will use the following well-known construction (see [BG] and [Pi], 3.3.2). For a pair of points x, y in the hyperbolic upper half-plane H, write ℓ(x, y) for the oriented geodesic joining x to y. Let Ω = dx ∧ dy/y^{2} be the hyperbolic area form.

Definition 3.2.A. A one-form α on H is bounded if there is some C > 0 so that |\frac{dα}{Ω}| ≤ C.

Given a bounded G-invariant one-form on H and a base-point x ∈ H, set

r_{x}(g) = r_{x}^{α}(g) := ∫_{ℓ(x,gx)} α

Lemma 3.2.B. If α is a bounded G-invariant one-form on H then

(i) r_{x} is a quasi-morphism.

(ii) |r_{x} − r_{y}| ≤ C_{α}.

Proof: 1) Let g, h ∈ G and consider

δr_{x}(g, h) := r_{x}(g) + r_{x}(h) − r_{x}(gh) ,

which we want to show is bounded. By G-invariance of α, we have

r_{x}(h) := ∫_{ℓ(x,hx)} α = ∫_{ℓ(gx,ghx)} α

Therefore

δr_{x}(g, h) = (∫_{ℓ(x,gx)} + ∫_{ℓ(gx,ghx)} − ∫_{ℓ(x,ghx)}) α = ∫_{∂T} α

is the integral around the oriented boundary of the geodesic triangle T with vertices at x, gx and ghx. By Stokes’ theorem, this equals the integral of dα on T, and thus if |dα| ≤ CΩ then

|δr_{x}(g, h)| = |∫_{T} dα| ≤ C |∫_{T} Ω| = C · area(T) .

Since the area of a geodesic triangle in the hyperbolic plane is at most π, we find that |δr_{x}(g, h)| ≤ πC is bounded and thus r_{x} is a quasi-morphism.

2) To see independence of r_{x} on the base-point up to a bounded quantity, consider the integral of α over the boundary of the geodesic parallelogram P with vertices at x, gx, gy and y, and again use Stokes’ theorem:

(∫_{ℓ(x,gx)} + ∫_{ℓ(gx,gy)} + ∫_{ℓ(gy,y)} + ∫_{ℓ(y,x)}) α = ∫_{P} dα = O(1) .

By G-invariance of α, we have

∫_{ℓ(gx,gy)} α = ∫_{ℓ(x,y)} α = − ∫_{ℓ(y,x)} α

and so we find

|r_{x}(g) − r_{y}(g)| = |∫_{ℓ(x,gx)} + ∫_{ℓ(gy,y)}| = |∫_{P} dα| ≤ 2πC

since the hyperbolic area of P is at most 2π. □

Since G is discrete, there are two kinds of elements of G with infinite order: hyperbolic and parabolic. The construction of r_{∞} is carried out separately for each of these two cases.

The hyperbolic case. It suffices to consider the case that g is a primitive hyperbolic element of G, that is we cannot write g = g_{1}^{k} for some g_{1} ∈ G and |k| ≥ 2. Thus we assume this to be the case from now.

We recall some facts from the geometry of discrete groups:

Any hyperbolic element leaves invariant a unique geodesic in H. Let L be the invariant geodesic for a primitive hyperbolic element g. The following is a standard fact:

Lemma 3.2.C.

(i) Suppose γ ∈ G, γL = L and γ preserves the orientation of L. Then γ = g^{k} for some integer k.

(ii) Suppose γ ∈ G, γL = L and γ reverses the orientation of L. Then γgγ^{−1} = g^{−1}.

Any discrete group G admits a fundamental region D for which the tessellation \{γ\overline{D} : γ ∈ G} of the upper half-plane H is locally finite, that is to say each compact subset of H intersects only finitely many of the translates γ\overline{D}. An example is the Dirichlet fundamental region of G [Be].

Lemma 3.2.D. There is a segment I ⊂ L and ϵ > 0 such that for every γ ∈ G, either γI ⊂ L or else dist(γI, L) > ϵ.

Proof: Let D be a locally finite fundamental domain of G which intersects L at an interior point. Write U_{δ} for the δ-neighborhood of L in the hyperbolic metric (the so-called “hypercycle domain”). Because D is locally finite, there are only a finite number of distinct translates γU_{δ}, γ ∈ G, which intersect the closure \overline{D} of D (see e.g. [Be, Theorem 9.2.8 (iii)]).

Let Y be the union of these finitely many translates of U_{δ}, which are distinct from U_{δ} itself. Decreasing δ, we can guarantee that Y ∩ L ∩ D ≠ L ∩ D.

Choose a segment I ⊂ L ∩ D such that I ∩ Y = ∅. Assume that for some γ ∈ G, γI ∩ U_{δ} ≠ ∅. Then I ∩ γ^{−1}U_{δ} ≠ ∅. But this means that γ^{−1}U_{δ} = U_{δ} due to our construction. This implies that γ^{−1}L = L so that γI ⊂ L. This proves the lemma (with ϵ = δ). □

Proof of the theorem in the hyperbolic case: Assume that g does not admit a timereversing symmetry. Then Lemmas 3.2.C and 3.2.D imply that if γ ≠ g^{k} for some k ∈ ℤ then γI is bounded away from L. Shrinking if necessary the segment I, we see that there exists its small neighborhood U such that γU is bounded away from U for all 1 ≠ γ ∈ G.

Choose a one-form α_{0} in H such that α_{0} has compact support, contained in U and

∫_{I} α_{0} > 0.

Let α be the extension of α_{0} by periodicity to the translates ∪_{γ∈G}γU. Note that α is a bounded one-form.

Now fix a point x ∈ L ∩ D such that U ∩ L lies in the interior of the geodesic segment ℓ(x, gx), and let r = r_{x}^{α} be the quasi-morphism constructed above. Clearly r(g^{n}) = ∫_{ℓ(x,g^{n}x)} α = n ∫_{I} α_{0} and so r_{∞}(g) = lim r(g^{n})/n = ∫_{I} α_{0} > 0. This completes the proof.

The parabolic case. Let h ∈ G be a parabolic element, which as in the hyperbolic case we may assume is primitive. Recall that parabolic elements are never conjugate to their inverses already in P SL(2, ℝ). Let L be an h-invariant horocycle and let U_{L} be a horocyclic domain in H, that is a neighborhood of the cusp fixed by h. For instance, if ∞ is a cusp for G and h(z) = z + 1 and we can take L = {y = C} and U_{L} = {y ≥ C}. The following is well-known:

Lemma 3.2.E. One can choose L so that γU_{L} ∩ U_{L} = ∅ for all γ ∈ G with γ ≠ h^{k}.

Proof: We may use a normal form for h and so assume that the cusp is at ∞ and that h(z) = z + 1. Then if γ ∈ G does not fix the cusp ∞, then γ = ( \begin{matrix}a & b \\ c & d \end{matrix}) with |c| ≥ 1 (see e.g. [Be, Proof of 9.2.8 (ii)]). In that case Im(γz) = Im(z)/|cz + d|^{2} ≤ 1/y. Thus the horocycles L = {y = C} for C > 1 satisfy the conditions of the Lemma. □

Proof of the theorem in the parabolic case: For simplicity we assume that h(z) = z+1, z = x + iy. Take L = {y = 2} and let u(y) be a smooth cutoff function, with u(y) ≡ 1 for y ≥ 3, and u(y) ≡ 0 if y ≤ 2.5. Set α_{0} = u(y)dx.

Note that α_{0} is bounded on H, since dα_{0} ≡ 0 for y ∉ (2.5, 3), while for 2.5 < y < 3 we have dα_{0} = u′(y)dy ∧ dx and comparing with Ω = y^{−2}dx ∧ dy gives |dα_{0}/Ω| = y^{2}u′(y) is bounded. Moreover α_{0} is supported in U_{L} = {y ≥ 2}.

Now let α = ∑_{γ∈G/⟨h⟩} γ^{∗}α_{0} be the periodization of α_{0}. The translates γ^{∗}α_{0} are supported in distinct translates γU_{L} for distinct γ modulo translates by powers of h. Thus we get a G-invariant, bounded one-form on H which equals dx on {y ≥ 3}.

Choose z with Im(z) = 3 and consider the quasi-morphism r = r_{z}^{α}. Then h^{n}(z) = z + n and clearly ∫_{ℓ(z,h^{n}z)} α = n. Thus r_{∞}(h) = 1 > 0 as required. This completes the proof of Theorem 1.5.D. □

Remark 3.2.F.The phenomenon described in Theorem 1.5.D holds true in a more general context of Gromov hyperbolic groups. In fact, if G is a non-elementary Gromov hyperbolic group then for every g ∈ G one has the following alternative. Either some positive power of g is conjugate to its inverse, or G admits a homogeneous quasi-morphism which is positive on g. This follows with minor extra efforts from a work by Epstein and Fujiwara (see [EF], proof of Lemma 3.5). Another natural generalization of 1.5.D is as follows. One asks whether there exists a quasi- morphism which attains prescribed values on a given finite subset of the group G. A solution of this problem for discrete subgroups of P SL(2, ℝ) as well as an application to stable mixing of linear maps of the 2-torus will be presented in a forthcoming paper [PR].

APPENDIX: The continuous case

This appendix to §3 is devoted to discussion of the continuous case, that is we take G to be P SL(2, ℝ) acting on P SL(2, ℝ)/Γ, where Γ is a lattice. We will take the subgroup

h^{t} = ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} )

which gives the horocycle flow on X. Our problem is (cf. 1.8.C above): Is the horocycle flow on P SL(2, ℝ)/Γ stably mixing?

3.3. Quasi-mixing.

First of all, let us relax the mixing property as follows.

Definition 3.3.A. A sequential system f_{∗} acting on a measure space (X, μ) by measurepreserving automorphisms is called quasi-mixing if there exists a sequence of positive integers i_{k} → +∞ such that for any L^{2}-functions F and H on X

∫_{X} F (f^{(i_{k})}x)H(x)dμ ⟶ ∫_{X} F (x)dμ ∫_{X} H(x)dμ

when k → ∞. That is, the subsequence {f^{(i_{k})}} is mixing.

Let Γ ⊂ P SL(2, ℝ) be a lattice. Consider the left action of P SL(2, ℝ) on the space

P SL(2, ℝ)/Γ endowed with the Haar measure. Write h^{t} = (\begin{matrix}1 & t \\ 0 & 1 \end{matrix}) for the horocycle flow,

and let ϕ_{∗} = {ϕ_{i}} be an arbitrary sequence of kicks from P SL(2, ℝ). Denote by QM(ϕ_{∗}) the set of those periods τ ∈ (0; +∞) for which the kicked system {ϕ_{i}h^{τ}} is quasi-mixing.

From general considerations, it follows that a quasi-mixing sequence is unbounded (that is has non-compact closure in P SL(2, ℝ)). It follows from the Howe-Moore theorem 1.5.A that the converse is also true.

Question 3.3.B. Is it true that for every sequence ϕ_{∗} the set QM(ϕ_{∗}) has “large measure”?

Write ϕ_{i} = (\begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix}). We give a partial affirmative answer to Question 3.3.B in terms of the

sequence {c_{i}}.

Theorem 3.3.C. If c_{i} = 0 for all i then the set (0; +∞)\QM(ϕ_{∗}) contains at most 1 point.

Theorem 3.3.D. Assume that c_{i} ≠ 0 for all i and the sequence \frac{1}{n} ∑_{i=1}^{n} log |c_{i}| is bounded

from below. Then the set (0; +∞)\QM(ϕ_{∗}) has finite measure.

We prove these theorems in 3.5, 3.6 below.

Interestingly enough, Theorems 3.3.C and 3.3.D handle two opposite cases: when all c_{i} vanish, and when all c_{i} are bounded away from 0. At present it is unclear how to attack the intermediate situation.

Remark 3.3.E. Let M(ϕ_{∗}) be the set of those periods τ for which the kicked system is mixing (equivalently, goes to infinity in P SL(2, ℝ)). In general, one cannot hope that this set has large measure. Here is an example which in fact reflects geometry of real numbers and has nothing to do with the Möbius group. We will produce a sequence of kicks of the form h^{β_{i}}, where the sequence {β_{i}} is chosen as follows. It is not hard to exhibit a sequence of intervals

I_{k} = [r_{k}; r_{k} + \frac{1}{k}] ⊂ [0; +∞), k ∈ ℕ

which cover every non-negative real number infinitely many times ^{11}.

Put now β_{k} = (k − 1)r_{k−1} − kr_{k}, where r_{0} = 0. One calculates that the evolution of the kicked system is given by f^{(k)}(τ) = h^{k(τ−r_{k})}. Pick up any positive real τ. Note that τ ∈ I_{k} if and only if k(τ − r_{k}) ∈ [0; 1], and thus the last inclusion holds true for infinite number of k due to our choice of intervals I_{k}. We conclude that for every value of τ the evolution of the kicked system does not go to infinity hence the system is not mixing. We still have no answer to the following question.

Question 3.3.F. Assume that the sequence of kicks ϕ_{∗} = {ϕ_{i}} has compact closure in P SL(2, ℝ). Is it true that the set M(ϕ_{∗}) has large measure?

3.4. A link to discrete Schrödinger equation.

Question 3.3.B turns out to be nontrivial even in the case when the kicks ϕ_{i} have a very simple form

ϕ_{i} = (\begin{matrix}1 & 0 \\ c_{i} & 1 \end{matrix}) ,

that is ϕ_{i}’s are time-c_{i}-maps of the conjugate horocycle flow

(\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) h^{−t} (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

Fix τ > 0, and write

f^{(k)}(τ) = (\begin{matrix}α_{k} & β_{k} \\ γ_{k} & δ_{k} \end{matrix}) ,

where f^{(k)}(τ) is the evolution of the kicked system. A straightforward calculation shows that the matrix coefficients satisfy the following recursive relations:

\begin{cases}α_{k} = α_{k−1} + τγ_{k−1} \\ γ_{k} = γ_{k−1} + c_{k}α_{k} \end{cases} ,

\begin{cases}β_{k} = β_{k−1} + τδ_{k−1} \\ δ_{k} = δ_{k−1} + c_{k}β_{k} \end{cases} .

11Indeed, one first partitions the divergent series ∑ 1/k into infinitely many divergent subseries a_{m,n}, ∑_{n} a_{m,n} = ∞ for all m. To do this, first divide the sequence 1/k into consecutive blocks so that the sum of elements in each block is at least 1. Then by taking a bijection j : N × N → N one defines the m-th subsequence as the union of the j(m, n)-th blocks, n = 1, 2, . . ..

Now denote by s_{m,n} = ∑_{t≤n} a_{m,t} (s_{m,0} := 0) the partial sums of the m-th subsequence, and construct the sequence of intervals J_{m,n} = [s_{m,n−1}, s_{m,n}]. Then for each m ≥ 1, ∪_{n≥1}J_{m,n} = [0, ∞) and so we get a sequence of intervals I_{k} = [r_{k}, r_{k} + 1/k] which cover every point infinitely many times.

Both sequences {α_{k}} and {β_{k}} satisfy the second order difference equation

(3.4.A)  q_{k+1} − (2 + τc_{k})q_{k} + q_{k−1} = 0 k ≥ 1 .

Note that this is the discrete Schrödinger equation with a potential, which depends on the parameter τ. Every solution of 3.4.A is uniquely determined by the initial conditions q_{0} and q_{1}. We get the following result.

Proposition 3.4.B. The sequence {f^{(k)}(τ)}, k ∈ ℕ is bounded if and only if all the solutions of the Schrödinger equation (3.4.A) are bounded.

It is instructive to translate Proposition 3.4.B into the language of operator theory. Consider the space V of all real sequences q = (q_{1}, q_{2}, …). Define linear operators on V,

L : (q_{1}, q_{2}, …, q_{i}, …) ↦ (c_{1}q_{1}, c_{2}q_{2}, …, c_{i}q_{i}, …)

and

Δ_{u} : (q_{1}, q_{2}, …, q_{i}, …) ↦ (uq_{1} − q_{2}, −q_{1} + 2q_{2} − q_{3}, …, −q_{i−1} + 2q_{i} − q_{i+1}, …) .

Here u ∈ ℝ is a parameter, and the i-th coordinate of Δ_{u}q is simply the second difference of the sequence q for all i ≥ 2. Consider an operator K_{u,τ} = τL + Δ_{u}. Note that every vector q ∈ KerK_{u,τ} describes a solution of the Schrödinger equation 3.4.A with the boundary condition q_{0} = (2 − u)q_{1}.

Consider the subspace V_{b} ⊂ V consisting of all bounded sequences. With this notation the discussion above leads to the following statement.

Proposition 3.4.C. The sequence {f^{(k)}(τ)}, k ∈ ℕ is bounded if and only if KerK_{u,τ} ⊂ V_{b} for all u ∈ ℝ.

The next result is a version of Theorem 3.3.D above.

Proposition 3.4.D. Suppose that |c_{i}| ≥ ε > 0 for all i ∈ ℕ. Then there exists τ_{0} > 0 such that the sequence {f^{(k)}(τ)} is unbounded for all τ > τ_{0}.

Proof: Fix u ∈ ℝ. We claim that for τ large enough KerK_{u,τ} ∩V_{b} = {0}. Assume the claim. Since dim KerK_{u,τ} = 1 we get that KerK_{u,τ} is not contained in V_{b} for τ large enough. Thus the desired result follows from 3.4.C.

It remains to prove the claim. Endow the space V_{b} with the norm ∥q∥ = sup_{i} |q_{i}|. Our assumption on c_{i} implies that operator L is invertible, L^{−1}(V_{b}) = V_{b} and ∥L^{−1}∥ ≤ \frac{1}{ε}. Further, Δ_{u}(V_{b}) ⊂ V_{b}, and Δ_{u} is a bounded operator. Denote ∥Δ_{u}∥ = v. We have to solve the equation K_{u,τ}q = 0, q ∈ V_{b}. Note then that (τL+Δ_{u})q = 0, that is (1l+τ^{−1}L^{−1}Δ_{u})q = 0. Since ∥L^{−1}Δ_{u}∥ ≤ v/ε we see that the operator 1l + τ^{−1}L^{−1}Δ_{u} is invertible for τ > v/ε, and therefore q = 0. This proves the claim. □

The proof above illustrates the difficulty which one faces in the case when the coefficients c_{i} are allowed to approach arbitrarily close to 0. Indeed, the operator L^{−1}Δ_{u} becomes unbounded, and one loses control on the kernel of 1l + τ^{−1}L^{−1}Δ_{u} even for large values of τ.

Let us present two additional cases when one gets the affirmative answer to Question 3.3.B

assuming that ϕ_{i} = (\begin{matrix}1 & 0 \\ c_{i} & 1 \end{matrix}).

(3.4.E) c_{i} → 0 when i → ∞;

(3.4.F) all c_{i} are non-negative.

Indeed assume that c_{i} → 0. We write |ψ| for the Euclidean norm of a matrix ψ ∈ P SL(2, ℝ), |ψ| = \sqrt{trψψ^{∗}}. Let us show that the sequence |f^{(k)}(τ)| is unbounded for every τ > 0. Assume on the contrary that for some τ > 0 holds |f^{(k)}(τ)| ≤ K for all k ∈ ℕ. Since c_{i} → 0 there exists i, j > 0 such that

|ϕ_{i+j}h^{τ} ⋯ ϕ_{i+1}h^{τ}| ≥ 2K^{2} .

But

|ϕ_{i+j}h^{τ} ⋯ ϕ_{i+1}h^{τ}| = |f^{(i+j)}(τ) · (f^{(i)}(τ))^{−1}| ≤ K^{2} .

This contradiction proves the claim. ^{12} Assume now that all c_{i} are non-negative, and for some τ > 0 the sequence |f^{(k)}(τ)|

is bounded. Write f^{(k)}(τ) = (\begin{matrix}α_{k} & β_{k} \\ γ_{k} & δ_{k} \end{matrix}), and note that our assumption implies that the

matrix coefficients are non-negative and bounded above. The recursive relations listed in the beginning of this section show that the sequence {α_{k}}, {β_{k}}, {γ_{k}}, {δ_{k}} are non-decreasing,

and thus they converge to some values α_{∞}, β_{∞}, γ_{∞}, δ_{∞}. Since the matrix (\begin{matrix}α_{∞} & β_{∞} \\ γ_{∞} & δ_{∞} \end{matrix}) belongs

to P SL(2, ℝ) we have that either α_{∞} ≠ 0 or β_{∞} ≠ 0. Assume without loss of generality that α_{∞} ≠ 0. Since c_{k} = \frac{γ_{k}−γ_{k−1}}{α_{k}} we conclude that c_{k} → 0 when k → ∞, and we are in the case (3.4.E) considered above. This completes the analysis of (3.4.E) and (3.4.F).

3.5. Proof of Theorem 3.3.C.

Write ϕ_{k} = (\begin{matrix}a_{k} & b_{k} \\ 0 & a_{k}^{−1} \end{matrix}). Fix k > 0 and for m ≤ k denote ψ_{m} = ϕ_{k} · . . . · ϕ_{m}. Then the

evolution of the kicked systems can be written as follows:

(3.5.A)  f^{(k)}(τ) = ϕ_{k}h^{τ} · . . . · ϕ_{1}h^{τ} = ( ∏_{j=0}^{k−1} ψ_{k−j}h^{τ}ψ_{k−j}^{−1}) · ψ_{1} .

Note that if ψ = (\begin{matrix}u & v \\ 0 & u^{−1} \end{matrix}), then

(3.5.B)  ψh^{τ}ψ^{−1} = (\begin{matrix}1 & τu^{2} \\ 0 & 1 \end{matrix}) .

The matrix ψ_{1} has the form (\begin{matrix}a_{1} · . . . · a_{k} & w_{k} \\ 0 & a_{1} · . . . · a_{k} \end{matrix}), where w_{k} is some real number. Sub-

stituting this to (3.5.A) and using (3.5.B) we get the following expression for the evolution of the kicked system:

12This argument was suggested to us by D. Kazhdan.

(3.5.C)  f^{(k)}(τ) = (\begin{matrix}a_{1} · . . . · a_{k} & w_{k} + τz_{k} \\ 0 & a_{1}^{−1} · . . . · a_{k}^{−1} \end{matrix}) ,

where

z_{k} = \frac{∑_{i=1}^{k}(a_{k} · . . . · a_{i})^{2}}{a_{1} · . . . · a_{k}} .

Now assume that for some τ_{0}, the sequence {f^{(k)}(τ)}, k ∈ ℕ is bounded. Then there exist constants α > β > 0 such that β ≤ |a_{1} ⋯ a_{k}| ≤ α for all k (look at the diagonal terms of 3.5.C). Therefore, for every k and i, we have |a_{k} · . . . · a_{i}| ≥ βα^{−1} and thus

|z_{k}| ≥ β^{2}α^{−3}k .

Note that

f^{(k)}(τ) − f^{(k)}(τ_{0}) = (\begin{matrix}0 & r_{k}(τ) \\ 0 & 0 \end{matrix}) ,

where r_{k}(τ) = z_{k}(τ − τ_{0}). Since |r_{k}(τ)| ≥ β^{2}α^{−3}k|τ − τ_{0}|, we conclude that the sequence f^{(k)}(τ) is unbounded for every τ ≠ τ_{0}. This completes the proof. □

3.6. Proof of Theorem 3.3.D.

We start with the following

Lemma 3.6.A. Let p_{k} be a sequence of real polynomials of degree k with leading coefficients α_{k}. Suppose that |α_{k}| ≥ λ^{k} for some λ > 0. Then the set

Y = {τ ∈ ℝ| the sequence {p_{k}(τ)} is bounded}

has finite Lebesgue measure.

Proof of 3.3.D: Write ϕ_{k} = (\begin{matrix}a_{k} & b_{k} \\ c_{k} & d_{k} \end{matrix}), and put p_{k}(τ) = trace (f^{(k)}(τ)), where f^{(k)}(τ) =

ϕ_{k}h^{τ} · . . . · ϕ_{1}h^{τ}. It is easy to see that p_{k}(τ) is a degree k polynomial with the leading coefficient c_{1} · . . . · c_{k}. Our assumption on the sequence {c_{i}} implies that |c_{1} … c_{k}| ≥ λ^{k} for some λ > 0. Applying Lemma 3.6.A, we get the statement of the Theorem. □

Proof of 3.6.A: Take T > 0, and write Y_{T} = Y ∩ [−T; T]. The sequence \frac{1}{k}p_{k}(τ) converges to 0 when k → ∞ for every τ ∈ Y_{T}. Applying the Egorov theorem [FW] we get that there exists a subset Z_{T} ⊂ Y_{T} such that

measureZ_{T} ≥ \frac{1}{2}measureY_{T}

and the sequence \frac{1}{k}p_{k}(τ) converges uniformly on Z_{T}. In particular, there exists k_{0} > 0 such that |p_{k}(τ)| ≤ k for all τ ∈ Z_{T} and k ≥ k_{0}. Setting p̃_{k}(τ) = α_{k}^{−1}p_{k}(τ), this implies that

(3.6.B)  |p̃_{k}(τ)| ≤ k|α_{k}|^{−1} for all τ ∈ Z_{T} .

Note that p̃_{k}(τ) is a polynomial of degree k with the leading coefficient 1. A theorem due to Polya (see [T], 2.9.13) states that for any measurable subset Z ⊂ ℝ

(3.6.C) .  max_{τ∈Z} |p̃_{k}(τ)| ≥ 2 · (\frac{measure Z}{4} )^{k}

Substituting Z = Z_{T} and combining with (3.6.B) we get

k|α_{k}|^{−1} ≥ 2 · (\frac{measure Z_{T}}{4} )^{k} ≥ 2 · (\frac{measure Y_{T}}{8} )^{k} .

Since |α_{k}| ≥ λ^{k} we obtain that

k ≥ 2 · (\frac{measure Y_{T}}{8} · λ)^{k} .

This inequality holds for every k ≥ k_{0}, so

measure Y_{T} ≤ \frac{8}{λ} .

Since this is true for every T > 0 we conclude that

measure Y ≤ \frac{8}{λ} .

This completes the proof. □

4. Stable super-recurrence in Hamiltonian dynamics

In the present section we prove slightly more general versions of Theorems 1.6.A,B,C. Recall that these theorems provide a sufficient condition for kick stable energy conservation law and super-recurrence in terms of Hofer’s geometry of the group of Hamiltonian diffeomorphisms Ham(X, Ω). In practice, it is easier to perform measurements not on Ham(X, Ω) itself but on its universal cover. It turns out that these simpler measurements are powerful enough to detect stable super-recurrence.

At the very end of the section we prove Proposition 1.7.F stated in the Introduction.

4.1. Dynamical preliminaries

In this section we describe a link between quasi-integrals (see 1.1.C) and super-recurrence (see 1.1.D) in the context of general sequential systems. Let X be a compact topological space endowed with a Borel probability measure μ. Let f_{∗} = {f_{i}} be a sequential system which acts on X by μ-preserving homeomorphisms. Assume that f_{∗} is not strictly ergodic (see 1.1.B ). Then there exists a continuous function F on X with zero mean such that F ≢ 0 and

(4.1.A)  lim sup_{N→∞} max \frac{1}{N} ∑_{i=0}^{N−1} F ∘ f^{(i)} ≥ α max F

for some α ∈ (0; 1]. In this case we say that F is an α-quasi-integral of f_{∗}.

For an open subset A ⊂ X and N ∈ ℕ define the counting function ν_{N,A} : X → ℝ as follows. For x ∈ X put

ν_{N,A}(x) = #{i ∈ [0; N − 1] | f^{(i)}(x) ∈ A}.

Define the quantity

R(f_{∗}, A) = lim sup_{N→+∞} max_{x∈X} \frac{1}{N} ν_{N,A}(x).

For every ϵ > 0 there exist arbitrarily long finite pieces {x_{0} = x, ..., x_{N−1}} of trajectories of f_{∗} which visit A with the frequency at least R(f_{∗}, A) − ϵ. Clearly, A is super-recurrent if R(f_{∗}, A) is strictly bigger than μ(A).

It turns out that in some situations one can extract fairly explicit information on superrecurrent sets from quasi-integrals. Let F be an α-quasi-integral of f_{∗}. Put γ = |\frac{min F}{max F} |, and

for c ∈ (0; 1) define the set

A_{c} = {x ∈ X | F (x) ≥ c max F } .

Theorem 4.1.B. Suppose that γ < \frac{α^{2}}{4−4α}. Then for every c ∈ (0; 1) which satisfies

|c − \frac{α}{2}| < \frac{1}{2}\sqrt{a^{2} + 4αγ − 4γ}

the set A_{c} is super-recurrent for the system f_{∗}. Moreover, R(f_{∗}, A_{c}) − μ(A_{c}) ≥ \frac{α−c}{1−c} − \frac{γ}{c+γ}.

The proof is based on the following lemmas.

Lemma 4.1.C. For every c ∈ (0; α)

R(f_{∗}, A_{c}) ≥ \frac{α − c}{1 − c} .

Proof. Choose arbitrary ε > 0. Denote

I_{N} = \frac{1}{N} ∑_{i=0}^{N−1} F ∘ f^{(i)} .

There exists an arbitrarily large positive integer N such that for some x_{0} ∈ X

I_{N}(x_{0}) ≥ (α − ε) max F .

Write ν_{N} for the counting function ν_{N,A_{c}} defined in the beginning of this section. Clearly,

NI_{N}(x_{0}) ≤ c max F (N − ν_{N}(x_{0})) + max F · ν_{N}(x_{0}) .

Combining this with the previous inequality we get that

max_{x} \frac{ν_{N}(x)}{N} ≥ \frac{α − c − ε}{1 − c} .

Since this holds true for all ε > 0 and for an infinite sequence of positive values of N we conclude that

R(f_{∗}, A_{c}) = lim sup_{N→+∞} \frac{ν_{N}(x)}{N} ≥ \frac{α − c}{1 − c} .  □

Lemma 4.1.D. For every c ∈ (0; α)

μ(A_{c}) ≤ \frac{γ}{c + γ} .

Proof. Note that

0 = ∫_{X} F dμ = ∫_{A_{c}} F dμ + ∫_{X\A_{c}} F dμ ≥ c max F · μ(A_{c}) + min F · (1 − μ(A_{c})) .

Thus

μ(A_{c}) ≤ \frac{− min F}{c max F − min F} = \frac{γ}{c + γ} .  □

Proof of 4.1.B. The assumptions of the theorem guarantee that \frac{α−c}{1−c} > \frac{γ}{c+γ}. Applying 4.1.C and 4.1.D we get that

R(f_{∗}, A_{c}) − μ(A_{c}) ≥ \frac{α − c}{1 − c} − \frac{γ}{c + γ} > 0 .

This completes the proof. □

4.2 Geometric preliminaries

Let (X, Ω) be a closed symplectic manifold. For a smooth path q^{t}, t ∈ [a; b] of Hamiltonian diffeomorphisms of (X, Ω) set

length (q^{t}) = ∫_{a}^{b} max_{x∈X} Q(x, t)dt ,

where Q is the normalized Hamiltonian generating the path. Here we use the following normalization:

∫_{X} Q(x, t)dμ = 0

for all t ∈ [a, b]. Let (h^{t}) be a one parameter subgroup of Ham(X, Ω) generated by the time-independent Hamiltonian H. Define a function ℓ_{H} : [0; +∞) → [0; +∞) by

ℓ_{H}(s) = inf length (q^{t}) ,

where the infimum is taken over all paths (q^{t}), t ∈ [0; s] with the following properties:

• q_{0} = 1l, q^{s} = h^{s};

• the paths q^{t} and h^{t} are homotopic through smooth paths with fixed end points.

Clearly,

(4.2.A)  s max H ≥ ℓ_{H}(s) ≥ ρ(h^{s}) ,

where ρ is the positive part of Hofer’s norm defined in 1.6. It is easy to see that ℓ_{H}(s + t) ≤ ℓ_{H}(s) + ℓ_{H}(t) for all s, t > 0, so the limit

ℓ_{∞}(H) = lim_{S→∞} \frac{ℓ_{H}(s)}{s max H}

exists. This quantity always belongs to the unit segment [0; 1].

In a number of interesting situations one can find non-trivial lower bounds for ℓ_{∞}(H) using tools of modern symplectic topology. Here we present such a bound (see 4.2.D below) which was obtained in [P1],[P4]. Recall that a submanifold L ⊂ (X, Ω) is called Lagrangian if dim L = \frac{1}{2} dim X, and the symplectic form Ω vanishes on T L.

Definition 4.2.B. Let L ⊂ X be a closed Lagrangian submanifold. We say that L has the Lagrangian intersection property if L ∩ ϕ(L) ≠ ∅ for every Hamiltonian diffeomorphism ϕ ∈ Ham(X, Ω).

For example, an equator of the 2-sphere (that is a simple closed curve which divides the sphere into two discs of equal areas) clearly has the Lagrangian intersection property.

Consider the cylinder T^{∗}S^{1} endowed with coordinates r ∈ ℝ and t ∈ S^{1}. The standard symplectic form on T^{∗}S^{1} is written as dr ∧ dt. Denote by Z the zero section {r = 0}. For a symplectic manifold (X, Ω) consider the topological stabilization (X × T^{∗}S^{1}, Ω + dr ∧ dt). If L is a closed Lagrangian submanifold of X then L × Z is a closed Lagrangian submanifold of X × T^{∗}S^{1}.

Definition 4.2.C. Let L ⊂ X be a closed Lagrangian submanifold. We say that L has the stable Lagrangian intersection property if L × Z has the Lagrangian intersection property in X × T^{∗}S^{1}.

Remark: It is easily seen that stable Lagrangian intersection property implies the Lagrangian intersection property.

In many situations, one can detect the stable Lagrangian intersection property with the help of the Floer homology. Let us give two examples.

• A closed Lagrangian submanifold L ⊂ X with π_{2}(X, L) = 0 has stable Lagrangian intersection property;

• The equator of the 2-sphere, which trivially has the Lagrangian intersection property, can be shown to in fact have the stable Lagrangian intersection property.

We refer to [P1],[P4] for further details and references.

Theorem 4.2.D. [P1],[P4] Let (h^{t}) be a one parameter subgroup of Ham(X, Ω) generated by a normalized Hamiltonian function H. Assume that there exists a closed Lagrangian submanifold L ⊂ X with stable Lagrangian intersection property such that H(x) ≥ C > 0 for all x ∈ L. Then ℓ_{H}(s) ≥ Cs for all s > 0, and ℓ_{∞}(H) ≥ \frac{C}{max H}.

4.3. Detecting stable super-recurrence

Let (X, Ω) be a closed symplectic manifold. Let (h^{t}) be a one parameter subgroup of Ham(X, Ω) generated by a time-independent normalized Hamiltonian H. Take an arbitrary bounded sequence ϕ_{∗} = {ϕ_{i}} of Hamiltonian diffeomorphisms of (X, Ω), and consider the kicked system f_{∗}^{τ} = {ϕ_{i}h^{τ}}. Put

A_{c} = {x ∈ X | H(x) > c max H} ,

where c ∈ (0; 1).

Theorem 4.3.A. Suppose that ℓ_{H}(s) = s max H for all s > 0. Then for every c, α ∈ (0; 1) and ε > 0 there exists a subset P ⊂ (0; ∞) with the following properties:

• the set (0; +∞)\P has finite Lebesgue measure;

• for every τ ∈ P the Hamiltonian H is an α-quasi-integral of the kicked system f_{∗}^{τ};

• for every τ ∈ P the set A_{c} is super-recurrent for the kicked system f_{∗}^{τ} with R(f_{∗}^{τ}, A_{c}) > 1 − ε.

Proof of Theorem 1.2.D: Since the maximum set of H contains an equator, and the equator has the stable Lagrangian intersection property (see 4.2 above) we conclude from 4.2.D and 4.2.A that ℓ_{H}(s) = s max H for all s. The Theorem follows now from 4.3.A. □

The condition ℓ_{H}(s) = s max H is very restrictive. However our technique enables us to detect a weaker version of kick stable super-recurrence in a more general situation. Recall that the density of a subset P ⊂ (0; +∞) is

lim inf_{T →+∞} \frac{1}{T} measure (P ∩ (0; T]) .

Suppose that ℓ_{∞}(H) > 0 and choose any α ∈ (0; ℓ_{∞}(H)). Put

(4.3.B)  θ = \frac{ℓ_{∞}(H) − α}{1 − α}.

Theorem 4.3.C. There exists a subset P ⊂ (0; +∞) of density at least θ such that for every τ ∈ P the Hamiltonian H is an α-quasi-integral of the kicked system f_{∗}^{τ}.

Theorems 4.3.A and 4.3.C are proved in 4.5 below.

Proof of Theorem 1.6.A: The Theorem follows from 4.3.C. Indeed, θ → ℓ_{∞}(H) when α → 0. □

Let us describe an application of Theorem 4.3.C to stable super-recurrence. Set γ = |\frac{min H}{max H} |.

We assume that the following inequality holds:

(4.3.D)  γ(4 − 4ℓ_{∞}(H)) < ℓ_{∞}(H)^{2} .

Denote by P_{c,δ} (where c, δ ∈ (0; 1)) the set of those values of the period τ ∈ (0; +∞) such that the set A_{c} is super-recurrent for the kicked system f_{∗}^{τ} with R(f_{∗}^{τ}, A_{c}) − μ(A_{c}) > δ. Choose arbitrary α ∈ (0; ℓ_{∞}(H)) such that

γ < \frac{α^{2}}{4 − 4α}.

Pick any c ∈ (0; 1) which satisfies

(4.3.E)  |c − \frac{α}{2}| < \frac{1}{2}\sqrt{α^{2} + 4αγ − 4γ} .

Set

(4.3.F)  δ = \frac{α − c}{1 − c} − \frac{γ}{c + γ}

One concludes from (4.3.E) that δ is positive. The next result follows immediately from 4.3.C and 4.1.B. Here we assume (4.3.D), and define θ by (4.3.B).

Theorem 4.3.G. The density of P_{c,δ} is greater or equal to θ.

Proof of Theorem 1.6.B. It follows from 4.2.A that ℓ_{∞}(H) = 1. Thus assumption 4.3.D holds. Put c = ϵ and choose α so close to 1 that 4.3.E holds. Note that θ = 1. Define δ by 4.3.F. It follows from 4.3.G that A_{c} is super-recurrent when τ belongs to the set P_{c,δ} of density 1. □

Proof of Theorem 1.6.C. It follows from (4.2.A) that ℓ_{∞}(H) ≥ 0.9. Further, it is given that γ = 1. Thus assumption 4.3.D holds. Choose α = 0.83, and c = 0.4. Note that the inequality (4.3.E) is satisfied and θ = \frac{0.9−0.83}{1−0.83} > 0.4. Theorem follows now from 4.3.G. □

4.4. A geometric inequality

The main ingredient of our approach to Theorems 4.3.A and 4.3.C is the following upper bound on the function ℓ_{H}(s). Let (h^{t}) be a one parameter subgroup of Ham(X, Ω) generated by a time-independent Hamiltonian H with zero mean. Take an arbitrary sequence {ϕ_{i}} of Hamiltonian diffeomorphisms. Denote by f^{(i)}(τ) the evolution of the kicked system, that is

f^{(i)}(τ) = ϕ_{i}h^{τ}ϕ_{i−1}h^{τ} · . . . · ϕ_{1}h^{τ} ,

where i ≥ 1, τ ∈ (0; ∞). Put f^{(0)}(τ) ≡ 1l. We write ρ̄ for Hofer’s norm defined 1.2.C.

Theorem 4.4.A. For every N ∈ ℕ and T > 0 holds

ℓ_{H}(NT) ≤ ∫_{0}^{T} max_{X} ∑_{i=0}^{N−1} H ∘ f^{(i)}(t)dt + 2 ∑_{i=1}^{N−1} ρ̄(ϕ_{i}) .

Proof: The proof is divided into several steps.

1) Decompose h^{NT} = A_{N,T} B_{N,T}, where

A_{N,T} = h^{T} · ∏_{i=1}^{N−1} ψ_{i}h^{T} ψ_{i}^{−1} ,

with

ψ_{i} = ψ_{i,N} = ϕ_{N−1} · . . . · ϕ_{N−i} , i = 1, …, N − 1

and

B_{N,T} = ϕ_{N−1} · . . . · ϕ_{1} ∏_{j=1}^{N−1} h^{−jT} ϕ_{j}^{−1}h^{jT} .

Take ε > 0, and choose paths ϕ_{i}^{(s)}, s ∈ [0; T] of Hamiltonian diffeomorphisms which join 1l with ϕ_{i} so that the lengths of the paths ϕ_{i}^{(s)} and (ϕ_{i}^{(s)})^{−1} do not exceed ρ̄(ϕ_{i}) + ε for all i. Consider the following paths of Hamiltonian diffeomorphisms defined for s ∈ [0; T]:

c_{s} = h^{Ns}

a_{s} = h^{s} ∏_{i=1}^{N−1} ψ_{i}h^{s}ψ_{i}^{−1}

and

b_{s} = ϕ_{N−1}^{(s)} ⋯ ϕ_{1}^{(s)} ∏_{j=1}^{N−1} h^{−jT} (ϕ_{j}^{(s)})^{−1}h^{jT} .

The paths {c_{s}} and {a_{s}b_{s}} join 1l with h^{NT}.

2) We claim that the paths {c_{s}} and {a_{s}b_{s}} are homotopic with fixed endpoints. Indeed take the parameter of homotopy u ∈ [0; 1] and write

a_{s,u} = h^{s} ∏_{i=1}^{N−1} ψ_{i}^{(u)}h^{s}(ψ_{i}^{(u)})^{−1} ,

where ψ_{i}^{(u)} = ϕ_{N−1}^{(T u)} ⋯ ϕ_{N−i}^{(T u)}. Set

b_{s,u} = ϕ_{N−1}^{(su)} ⋯ ϕ_{1}^{(su)} ∏_{j=1}^{N−1} h^{−jT} (ϕ_{j}^{(su)})^{−1}h^{jT} .

The required homotopy is given by d_{s,u} = a_{s,u}b_{s,u} where s ∈ [0; T] and u ∈ [0; 1].

3) It follows from the definition of the function ℓ_{H} that ℓ_{H}(NT) ≤ length(a_{s}b_{s})_{s∈[0;T]}. But length (a_{s}b_{s}) ≤ length(a_{s}) + length(b_{s}), and

length(b_{s}) ≤ 2 ∑_{j=1}^{N−1} ρ̄(ϕ_{j}) + 2(N − 1)ε

(one uses here that Hofer’s norm ρ̄ is bi-invariant [H], [P4]). We conclude that

(4.4.B) ℓ_{H}(NT) ≤ length(a_{s})_{s∈[0;T]}+

+2 ∑_{j=1}^{N−1} ρ̄(ϕ_{j}) + 2(N − 1)ε.

4) Denote by F̃(x, s) = F̃_{s}(x) the normalized Hamiltonian function generating (a_{s}). In order to calculate F̃, we use the following product formula: Let (p_{s}), (q_{s}), s ∈ [0; T] be two Hamiltonian flows generated by normalized Hamiltonians P (x, s) and Q(x, s). Then the product (p_{s}q_{s}) is a Hamiltonian flow generated by normalized Hamiltonian P (x, s) + Q(p_{s}^{−1}x, s). In particular, for a given Hamiltonian diffeomorphism ψ the path (ψh^{s}ψ^{−1}) is generated by Hamiltonian H ∘ ψ^{−1}.

Applying the product formula we get that

F̃_{s} = H + ∑_{i=1}^{N−1} H ∘ ψ_{i}^{−1} ∘ ( ∏_{j=0}^{i−1} ψ_{j}h^{s}ψ_{j}^{−1})^{−1} ,

where ψ_{0} = 1l. Using that ψ_{k−1}^{−1}ψ_{k} = ϕ_{N−k} we can rewrite this as

F̃_{s} = H + ∑_{i=1}^{N−1} H ∘ ( ∏_{j=1}^{i} h^{s}ϕ_{N−j})^{−1} .

Introduce a new function

F_{s} = F̃_{s} ∘ ( ∏_{j=1}^{N−1} h^{s}ϕ_{N−j}) ∘ h^{s} .

Then

F_{s} = H ∘ h^{s} ∘ ( ∏_{j=1}^{N−1} ϕ_{N−j}h^{s}) +

+ ∑_{i=1}^{N−2} H ∘ ( ∏_{j=i+1}^{N−1} h^{s}ϕ_{N−j}) ∘ h^{s} + H ∘ h^{s} .

The energy conservation law implies that H ∘ h^{s} = H. Thus

F_{s} = H ∘ ∏_{j=1}^{N−1} ϕ_{N−j}h^{s} +

+ ∑_{i=1}^{N−2} H ∘ ϕ_{N−i−1}h_{s}ϕ_{N−i−2}h_{s} ⋯ ϕ_{1}h_{s} + H .

The last expression can be rewritten in terms of the kicked system:

F_{s} = ∑_{k=0}^{N−1} H ∘ f^{(k)}(s) .

Since max_{X} F_{s} = max_{X} F̃_{s} for all s, we get that

length(a_{s})_{s∈[0;T]} = ∫_{0}^{T} max_{X} F̃_{s}ds =

= ∫_{0}^{T} max_{X} ∑_{k=0}^{N−1} H ∘ f^{(k)}(s)ds .

Substituting this expression into (4.4.B) we get the desired inequality 4.4.A. This completes the proof.

4.5 Proof of main theorems

Proof of Theorem 4.3.C. Set H_{N}(t) = \frac{1}{N} max_{X} ∑_{i=0}^{N−1} H ∘ f^{(i)}(t). Let P be the set of those

t ∈ (0; ∞) for which the inequality H_{N}(t) ≥ α max H holds for an infinite sequence of positive integers N. We have to show that the density of P is at least θ = \frac{ℓ_{∞}(H)−α}{1−α}.

Since the sequence {ϕ_{i}} is bounded, there exists u > 0 such that 2ρ̄(ϕ_{i}) ≤ u for all i. Fix κ > 0, and abbreviate ℓ = ℓ_{∞}(H). There exist T_{0} > 0, N_{0} > 0 such that for all N > N_{0}, T > T_{0}

ℓ_{H}(NT) ≥ NT (ℓ − κ) max H .

The geometric inequality 4.4.A yields

ℓ_{H}(NT) ≤ N ∫_{0}^{T} H_{N}(t)dt + (N − 1)u .

Therefore

NT (ℓ − κ) max H ≤ N ∫_{0}^{T} H_{N}(t)dt + Nu .

Increasing T_{0} we can assume that u ≤ κT max H, so we get

(4.5.A)  ∫_{0}^{T} H_{N}(t)dt ≥ (ℓ − 2κ)T max H .

Consider the set

Q_{N,T} = {t ∈ [0; T] | H_{N}(t) < α · max H} .

Clearly,

∫_{0}^{T} H_{N}(t)dt = ∫_{Q_{N,T}} H_{N}(t)dt + ∫_{[0;T]\Q_{N,T}} H_{N}(t)dt ≤

≤ α max H · measure(Q_{N,T}) + max H · (T − measure(Q_{N,T})) .

Combining this with 4.5.A we get that

(4.5.B)  \frac{measure(Q_{N,T})}{T} ≤ \frac{1 − ℓ + 2κ}{1 − α}

Clearly, [0; T]\P = ⋃_{k=1}^{∞} ⋂_{N=k}^{∞} Q_{N,T}, so in view of (4.5.B)

\frac{1}{T} measure(P ∩ [0; T]) ≥ \frac{ℓ − α − 2κ}{1 − α}

for T > T_{0}. Thus density(P) ≥ θ = \frac{ℓ−α−2κ}{1−α}. Since κ can be chosen arbitrary small, this completes the proof of the theorem. □

Proof of Theorem 4.3.A The proof is analogous to the one of 4.3.C. Take α ∈ (0; 1) and define H_{N}(t), u and Q_{N,T} as above. Since ℓ_{H}(NT) = NT max H due to our assumption, and

ℓ_{H}(NT) ≤ N ∫_{0}^{T} H_{N}(t)dt + Nu

in view of 4.4.A, we get that

∫_{0}^{T} H_{N}(t)dt ≥ T max H − u .

Therefore

measure(Q_{N,T}) ≤ \frac{u}{(1 − α) max H} .

The set Q = ⋃_{T} ⋃_{k=1}^{∞} ⋂_{N=k}^{∞} Q_{N,T} has finite measure. Consider its complement P = (0; +∞)\Q. For every τ ∈ P function H is an α-quasi-integral of the kicked system f_{∗}^{τ}.

Assume without loss of generality that α is sufficiently close to 1 so that the following hold:

• |c − \frac{α}{2} | < \frac{1}{2} \sqrt{α^{2} + 4αγ − 4γ};

• \frac{α−c}{1−c} ≥ 1 − ε , \frac{α−c}{1−c} > \frac{γ}{c+γ}.

Here γ stands for |\frac{min H}{max H} |, and ε is given in the formulation of 4.3.A. It follows from 4.1.B

that the set A_{c} = {x ∈ X | H(x) ≥ c max H} is super-recurrent for f_{∗}^{τ}. Moreover, Lemma 4.1.C guarantees that R(f_{∗}^{τ}, A_{c}) ≥ 1 − ε. This completes proof. □

4.6 Creating random behaviour

Here we prove Proposition 1.7.F of the introduction. Let (X, Ω) be a closed symplectic manifold such that Ham(X, Ω) is C^{∞}-closed in Diff(X). Note that under this assumption there exists a bounded strictly ergodic sequence ψ_{∗} = {ψ_{i}} of Hamiltonian diffeomorphisms. This is an immediate consequence of [P3], Th. 1.2.A.

Let θ be a time reversing symmetry for (h^{t}), that is θh^{t}θ^{−1} = h^{−t} for all t ∈ ℝ. Define a sequence of kicks {ϕ_{i}} as follows. Set ϕ_{i} = θ^{−1} when i is odd, and ϕ_{i} = ψ_{k}θ when i is even and equals 2k. We claim that for every τ > 0 the kicked system f_{∗}^{τ} = {ϕ_{i}h^{τ}} is strictly ergodic. Indeed, note that f^{(i)}(τ) = ϕ_{i}h^{τ} · . . . ϕ_{1}h^{τ} equals ψ^{(k)} for i = 2k, and θ^{−1}h^{τ}ψ^{(k)} for

i = 2k + 1. For every continuous function F on X set I_{N} = ∑_{i=0}^{N−1} F ∘ f^{(i)}(τ). We see that for

N = 2k, I_{N} = ∑_{i=0}^{k−1} F ∘ ψ^{(i)} + ∑_{i=0}^{k−1}(F ∘ θ^{−1}h^{τ}) ∘ ψ^{(i)}. Thus the uniform limit lim_{N→∞} \frac{1}{N} I_{N} exists

and equals \frac{1}{2}( ∫_{X} F dμ + ∫_{X} F ∘ θ^{−1}h^{τ}dμ) = ∫_{X} F dμ. Therefore f_{∗}^{τ} is strictly ergodic for every

τ. This completes the proof. □

Example 4.6.A We conclude this section with an example in the spirit of 1.7.G and 1.7.H. It shows that the phenomenon presented in Theorem 4.3.A is a purely Hamiltonian one and may disappear when one allows kicks which are symplectic but not necessarily Hamiltonian diffeomorphisms of (X, Ω). Consider the 2-torus 𝕋^{2} = ℝ^{2}(x, y)/ℤ^{2} endowed with the symplectic form Ω = dx ∧ dy. Consider a one parameter subgroup (h^{t}) of Hamiltonian diffeomorphisms given by

h^{t}(x, y) = (x, y − t sin 2πx) .

It is generated by a normalized Hamiltonian function H(x, y) = \frac{1}{2π} cos 2πx, which attains its maximal value on a non-contractible curve L = {x = 0}. Since L has stable Lagrangian intersection property (see 4.2.C above) it follows from Theorem 4.2.D that ℓ_{H}(s) ≡ s max H. Thus Theorem 4.3.A implies stable super-recurrence. On the other hand the shift θ : (x, y) ↦ (x + \frac{1}{2}, y) is a time reversing symmetry of (h^{t}). Note that θ is a symplectic, but not a Hamiltonian diffeomorphism. Our proof above shows that there exists a sequence of symplectomorphisms {ϕ_{i}} such that the kicked system {ϕ_{i}h^{τ}} is strictly ergodic for each value of τ.

References

[BR] M. Baake and J.A.G. Roberts, Reversing symmetry group of GL(2, ℤ) and P GL(2, ℤ) matrices with connection to cat maps and trace maps, J. Phys. A: Math. Gen. 30(1997), 1549-1573.

[BG] J. Barge and E. Ghys, Surfaces et cohomologie bornee, Invent. Math. 92 (1988), 509-526.

[Ba] C. Bavard, Longueur stable des commutateurs, Enseign. Math. (2) 37 (1991), 109– 150.

[Be] A.F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, 1983.

[BB] D. Berend and V. Bergelson, Ergodic and mixing sequences of transformations, Ergodic Th. Dynam. Syst. 4(1984), 353-366.

[Br] R. Brooks, Some remarks on bounded cohomology, in Riemann Surfaces and Related Topics, I. Kra and B. Maskit eds., Princeton University Press, 1980, pp. 53-63.

[BM1] M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups, Journal of the European Math. Soc. 1 (1999), 199-235 and 338.

[BM2] M. Burger and N. Monod, Continuous bounded cohomology and its applications preprint, January 2000.

[EP] Y. Eliashberg and L. Polterovich, Bi-invariant metrics on the group of Hamiltonian diffeomorphisms, Internat.J.Math. 4 (1993), 727-738.

[EF] D. Epstein and K. Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), 1275-1289.

[FP] M. Field and W. Parry, Stable ergodicity of skew extensions by compact Lie groups, Topology 38 (1999), 167-187.

[FW] W. Filter and K. Weber, Integration Theory, Chapman & Hall, London, 1997.

[HKS] F. Haake, M. Kus and R. Scharf, Classical and quantum chaos for a kicked top, Zeitsch. Phys. B 65(1987), 381-395.

[Ha] M.B. Haeri, Chaos in Liouville’s equation, Phys. Rev. E 48(1993), 4215-4220.

[He] J.F. Heagy, A physical interpretation of the Henon map, Physica D 57(1992), 436-446.

[H] H. Hofer, On the topological properties of symplectic maps, Proc. Royal Soc. Edinburgh 115A (1990), 25-38.

[Ki] Y. Kifer, Random dynamics and its applications, in ”Proceedings of the ICM, Berlin, 1998”, Vol. II, pp. 809-818.

[KN] L. Kuipers and H. Niederreiter, Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience (John Wiley & Sons), New York-London-Sydney, 1974.

[LMP] F. Lalonde, D. McDuff and L. Polterovich, On the flux conjectures, in ”Geometry, Topology and Dynamics, ed F. Lalonde, Proceedings of the CRM 1995 Workshop in Montreal, the CRM Special Series of the AMS”, Vol.15, 1998, pp. 69-85.

[LR] J.S.W. Lamb and J.A.G. Roberts, Time-reversal symmetry in dynamical systems. A survey, Physica D, 112(1998), 1-39.

[Pi] J.-C. Picaud, Cohomologie bornee de surfaces et courants geodesiques, Bull. Soc. Math. France 125 (1997), 115-142.

[P1] L. Polterovich, Hofer’s diameter and Lagrangian intersections, Int.Math.Research Notices 4 (1998), 217-223.

[P2] L. Polterovich, Geometry on the group of Hamiltonian diffeomorphisms, in ”Proceedings of the ICM, Berlin, 1998”, Vol. II, pp. 401-410.

[P3] L. Polterovich, Hamiltonian loops from the ergodic point of view, Journal of the European Math. Soc. 1 (1999), 87 - 107.

[P4] L. Polterovich, Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser, 2001.

[PR] L. Polterovich and Z. Rudnick, Stable mixing for cat maps and quasi-morphisms of the modular group, preprint.

[S] G. Sell, Topological dynamics and ordinary differential equations. Van Nostrand Reinhold Mathematical Studies, No. 33. Van Nostrand Reinhold Co., London, 1971.

[T] A.F. Timan, Theory of approximation of functions of a real variable, International Series of Monographs in Pure and Applied Mathematics, Vol. 34, Pergamon Press, 1963.

[V] M. Viana, Dynamics: a probabilistic and geometric perspective, in ”Proceedings of the ICM, Berlin, 1998”, Vol. I, pp. 557-578.

[Ze] A. Zeghib, Isometry groups and geodesic foliations of Lorentz manifolds, Parts I and II, GAFA 9 (1999),775-854

[Zi] R.J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Math., Birkhäuser, 1984.