Consider a $C^{1+\epsilon}$ diffeomorphism $f$ having a uniformly hyperbolic compact invariant set $\Omega$, maximal invariant in some small neighbourhood of itself. The asymptotic exponential rate of escape from any small enough neighbourhood of $\Omega$ is given by the topological pressure of $-\log |J^u f|$ on $\Omega$ (Bowen–Ruelle in 1975). It has been conjectured (Eckmann–Ruelle in 1985) that this property, formulated in terms of escape from the support $\Omega$ of a (generalized Sinai–Ruelle–Bowen (SRB)) measure, using its entropy and positive Lyapunov exponents, holds more generally. We present a simple $C^\infty$ two-dimensional counterexample, constructed by a surgery using a Bowen-type hyperbolic saddle attractor as the basic plug.

Chaotic and ergodic properties are discussed for various subclasses of cylindric billiards. A common feature of the studied systems is that they satisfy a natural necessary condition for ergodicity and hyperbolicity, the so-called transitivity condition. The relation of our discussion to former results on hard ball systems is twofold. On the one hand, by slight adaptation of the proofs we may discuss hyperbolic and ergodic properties of 3 or 4 particles with (possibly restricted) hard ball interactions in any dimensions. On the other hand, a key tool in our investigations is a kind of connected path formula for cylindric billiards, which together with the conservation of momenta gives back, when applied to the special case of hard ball systems, the classical connected path formula.

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler–Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry–Mather theory of twist maps and the Hedlund–Morse–Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.

We study transfer operators over general subshifts of sequences of an infinite alphabet. We introduce a family of Banach spaces of functions satisfying a regularity condition and a decreasing condition. Under some assumptions on the transfer operator, we prove its continuity and quasi-compactness on these spaces. Under additional assumptions—existence of a conformal measure and topological mixing—we prove that its peripheral spectrum is reduced to one and that this eigenvalue is simple. We describe the consequences of these results in terms of existence and properties of invariant measures absolutely continuous with respect to the conformal measure. We also give some examples of contexts in which this setting can be used—expansive maps of the interval, statistical mechanics.

We consider an Abel equation $(*)$ $y^{\prime}=p(x)y^2+q(x)y^3$ with $p(x)$, $q(x)$ polynomials in $x$. A center condition for ($*$) (closely related to the classical center condition for polynomial vector fields on the plane) is that $y_0=y(0)\equiv y(1)$ for any solution $y(x)$ of ($*$). This condition is given by the vanishing of all the Taylor coefficients $v_k(1)$ in the development $y(x)=y_0+\sum^{\infty}_{k=2}v_k(x)y^k_0$. A new basis for the ideals $I_k=\{v_2,\dots,v_k\}$ has recently been produced, defined by a linear recurrence relation. Studying this recurrence relation, we connect center conditions with a representability of $P=\int p$ and $Q=\int q$ in a certain composition form (developing further some results of Alwash and Lloyd), and with a behavior of the moments $\int P^kq$. On this base, explicit center equations are obtained for small degrees of $p$ and $q$.

For a sequence $(c_n)$ of complex numbers we consider the quadratic polynomials $f_{c_n}(z):=z^2+c_n$ and the sequence $(F_n)$ of iterates $F_n:= f_{c_n} \circ \dotsb \circ f_{c_1}$. The Fatou set $\mathcal{F}_{(c_n)}$ is by definition the set of all $z \in \widehat{\mathbb{C}}$ such that $(F_n)$ is normal in some neighbourhood of $z$, while the complement of $\mathcal{F}_{(c_n)}$ is called the Julia set $\mathcal{J}_{(c_n)}$. The aim of this paper is to study the connectedness of the Julia set $\mathcal{J}_{(c_n)}$ provided that the sequence $(c_n)$ is bounded and randomly chosen. For example, we prove a necessary and sufficient condition for the connectedness of $\mathcal{J}_{(c_n)}$ which implies that $\mathcal{J}_{(c_n)}$ is connected if $|c_n| \le \frac{1}{4}$, while it is almost surely disconnected if $|c_n| \le \delta$ for some $\delta>\frac{1}{4}$.

Let $(E,{\cal A},\mu,T)$ be a dynamical system and let $\Phi$ be a function defined on $E$ with values in $\mathbb{R}^2$. We give a criterion, the central limit theorem along subsequences of positive density, for the recurrence of the corresponding ‘stationary walk’ defined as the cocycle $\big(\sum^{n-1}_{j=0}\Phi(T^jx)\big)_{n\geq1}$.

This criterion is satisfied by functions which are homologous to a martingale difference (a property which holds for regular functions in many systems). It can also be applied to the periodic Lorentz gas in the plane and shows recurrence for this model.

We show that the horocycle foliations on a compact $C^{\infty}$ (or even $C^{\omega}$) surface of non-positive curvature can fail to be Lipschitz, even if the curvature vanishes only along a single closed geodesic. We calculate the Hölder exponents of these foliations at vectors tangent to geodesics along which the curvature vanishes.

We present numerous examples of ways that a Bernoulli shift can behave relative to a family of factors. This shows the similarities between the properties which collections of ergodic transformations can have and the behavior of a Bernoulli shift relative to a collection of its factors. For example, we construct a family of factors of a Bernoulli shift which have the same entropy, and any extension of one of these factors has more entropy, yet no two of these factors sit the same. This is the relative analog of Ornstein and Shields uncountable collection of nonisomorphic $K$ transformations of the same entropy. We are able to construct relative analogs of almost all the zero entropy counter-examples constructed by Rudolph (1979), as well as the $K$ counterexamples constructed by Hoffman (1997). This paper provides a solution to a problem posed by Ornstein (1975).

This paper is concerned with the dynamics of transcendental entire functions. Let $f(z)$ be a transcendental entire function. We shall study the boundedness of the components of the Fatou set $F(f)$ under some restrictions on the growth of the function. This relates to a problem due to Baker in 1981.

First, we show that there exists a sequence $(a_n)$ of integers which is a good averaging sequence in $L^2$ for the pointwise ergodic theorem and satisfies $$ \frac{a_{n+1}}{a_n}>e^{(\log n)^{-1-\epsilon}} $$ for $n>n(\epsilon)$. This should be contrasted with an earlier result of ours which says that if a sequence $(a_n)$ of integers (or real numbers) satisfies $$ \frac{a_{n+1}}{a_n}>e^{(\log n)^{-\frac{1}{2}+\epsilon}} $$ for some positive $\epsilon$, then it is a bad averaging sequence in $L^2$ for the pointwise ergodic theorem.

Another result of the paper says that if we select each integer $n$ with probability $1/n$ into a random sequence, then, with probability 1, the random sequence is a bad averaging sequence for the mean ergodic theorem. This result should be contrasted with Bourgain's result which says that if we select each integer $n$ with probability $\sigma_n$ into a random sequence, where the sequence $(\sigma_n)$ is decreasing and satisfies $$ \lim_{t\to\infty}\frac{\sum_{n\le t}\sigma_n}{\log t}=\infty, $$ then, with probability 1, the random sequence is a good averaging sequence for the mean ergodic theorem.

Cocycles of $Z^m$ actions on compact metric spaces can be used to construct $R^m$ actions or flows, called suspension flows. A suspension provides a higher-dimensional analog to the familiar flow under a function and we look to this construction as a way of generating interesting $R^m$ flows. Even more importantly, an $R^m$ flow with a free dense orbit has an almost one-to-one extension which is a suspension [6] and thus suspensions can be used to model general $R^m$ flows. In this paper we examine the sensitivity of the suspension construction to small perturbations in the cocycle. Theorem 4.7 establishes the fact that two cocycles that are sufficiently close yield suspensions that are isomorphic up to a time change.

We prove the following result. Let $G$ be a countable discrete group with finite conjugacy classes, and let $(X_n, n\in\mathbb Z)$ be a two-sided, strictly stationary sequence of $G$-valued random variables. Then $\mathscr T_\infty =\mathscr T_\infty ^*$, where $\mathscr T_\infty$ is the two-sided tail-sigma-field $\bigcap_{M\ge1}\sigma (X_m:|m|\ge M)$ of $(X_n)$ and $T_\infty ^*$ the tail-sigma-field $\bigcap_{M\ge0}\sigma (Y_{m,n}:m,n\ge M)$ of the random variables $(Y_{m,n}, m,n\ge0)$ defined as the products $Y_{m,n}=X_n\dots X_{-m}$. This statement generalises a number of results in the literature concerning tail triviality of two-sided random walks on certain discrete groups.

Using pressure formulas we compute the Hausdorff dimension of the basic set of ‘almost every’ ${\cal C}^{1+\alpha}$ horseshoe map in $\mathbb{R}}^3$ of the form $F(x,y,z)= (\gamma(x,z), \tau(y,z), \psi(z))$, where $|\psi'|>1$ and $0< |\gamma'_x|, |\tau'_y| < \frac{1}{2}$ on the basic set. Similar results are obtained for attractors of nonlinear ‘baker's maps’ in $\mathbb{R}}^3$.

We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.