We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is mixing and, in particular, the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, by the average rate at which typical points start to exhibit exponential growth of the derivative.

We consider analytic maps $f_j:D\to D$ of a domain D into itself and ask when does the sequence $f_1\circ\dotsb\circ f_n$ converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence $\{w_1,w_2,\dotsc\}$ in D whose values are not taken by any fj in D, and which is homogeneous in the sense that it comes within a fixed hyperbolic distance of any point of D. The situation for several complex variables is also discussed.

We study the existence, structure and topological entropy of almost automorphic arrays in symbolic lattice dynamical systems. In particular, we show that almost automorphic arrays with arbitrarily large entropy are typical in symbolic lattice dynamical systems. Applications to pattern formation and spatial chaos in infinite-dimensional lattice systems are considered and the construction of chaotic almost automorphic signals is discussed.

Let f be a generalized eigenfunction of an interval exchange transformation, T, on the unit interval, which satisfies the infinite distinct orbit condition (IDOC). We assume that the minimum spacing, $\epsilon_n(T)$, of the partition defined by Tn is of the order 1/n for infinitely many n. This assumption is generic. Given $\delta>0$ we prove that for sufficiently large n and for every interval J satisfying $\vert J \vert =\epsilon_n(T)$, there exists $x_0 \in J$ such that \[\vert \{ x \in J : \vert f(x) - f(x_0) \vert \geq \delta \} \vert< \delta \vert J \vert.\] This provides a specific sufficient generic diophantine condition for Veech's result [V2, Lemma 7.3]. Above $\vert \cdot\vert$ denotes the linear measure of the set. Moreover, if T is uniquely ergodic, then any bounded variation continuous f must be of the form $f(x)= \gamma \exp(2 \pi iax)$ almost everywhere, where $\gamma \in \mathbb{C}$ and $a \in \mathbb{R}$.

We consider the empirical distribution of a trajectory generic with respect to the absolutely continuous invariant measure of an expanding map of the interval. We prove the convergence of the suitably normalized fluctuations to a Gaussian bridge. One can then apply Dudley's theorem to build a Kolmogorov–Smirnov test. We also prove an upper and a lower bound on the tail fluctuation using a variant of Slepian's inequality.

We show that the metric of non-positively curved graph manifolds is determined by its geodesic flow. More precisely, we show that if the geodesic flows of two non-positively curved graph manifolds are C0 conjugate, then the spaces are isometric.

In this paper we consider the complexification of the Arnold standard family of circle maps given by $\widetilde F_{\alpha,\epsilon}(u)=ue^{i\alpha} e^{({\epsilon}/{2}) (u-{1}/{u})}$, with $\alpha=\alpha(\epsilon)$ chosen so that $\widetilde F_{\alpha(\epsilon),\epsilon}$ restricted to the unit circle has a prefixed rotation number $\theta$ belonging to the set of Brjuno numbers. In this case, it is known that $\widetilde F_{\alpha(\epsilon),\epsilon}$ is analytically linearizable if $\epsilon$ is small enough and so it has a Herman ring $\widetilde U_{\epsilon}$ around the unit circle. Using Yoccoz's estimates, one has that the size$\widetilde R_\epsilon$ of $\widetilde U_{\epsilon}$ (so that $\widetilde U_{\epsilon}$ is conformally equivalent to $\{u\in{\mathbb C}: 1/\widetilde R_\epsilon < |u| < \widetilde R_\epsilon\}$) goes to infinity as $\epsilon\to 0$, but one may ask for its asymptotic behavior.

We prove that $\widetilde R_\epsilon=({2}/{\epsilon})(R_0+\mathcal{O}(\epsilon\log\epsilon))$, where R0 is the conformal radius of the Siegel disk of the complex semistandard map $G(z)=ze^{i\omega}e^z$, where $\omega= 2\pi\theta$. In the proof we use a very explicit quasiconformal surgery construction to relate $\widetilde F_{\alpha(\epsilon),\epsilon}$ and G, and hyperbolic geometry to obtain the quantitative result.

Let V be a finite-dimensional vector space over $\mathbb{R}$and let $\Gamma \subset \mathrm{GL}(V)$ be a semigroup. We study the closed $\Gamma$-invariant subsets of V-\{0\} under the condition that the Zariski closure of $\Gamma$ is semi-simple. We use the results to show that, if $\Gamma\subset\mathrm{SL}(\mathbb{R}^d)$ acts on $\mathbb{T}^d$ by automorphisms, then the orbits of $\Gamma$ are finite or dense.

We study the growth of fibers of coverings of pinched negatively curved Riemannian manifolds. The applications include counting estimates for horoballs in the universal cover of geometrically finite manifolds with cusps. Continuing our work on diophantine approximation in negatively curved manifolds started in an earlier paper (Math. Zeit.241 (2002), 181–226), we prove a Khintchine–Sullivan-type theorem giving the Hausdorff measure of the geodesic lines starting from a cusp that are well approximated by the cusp returning ones.

In Blanchard et al (Topological complexity. Ergod. Th. & Dynam. Sys.20 (2000), 641–662), the authors introduced the notion of scattering and a weaker notion of 2-scattering. It is an open question whether the two notions are equivalent. The question is answered affirmatively in this paper. Using the complexity function of an open cover along some sequences of natural numbers, we characterize mild mixing, strong scattering and scattering. We show that mildly mixing (respectively strongly mixing) systems are disjoint from minimal uniformly rigid (respectively minimal rigid) systems.

Moreover, assuming minimality we show that a dynamical system is full-scattering (respectively mildly mixing or weakly mixing) if and only if it is strongly mixing (respectively IP*-transitive or $\mathcal{D}$-transitive), where $\mathcal{D}$ is the collection of subsets of $\mathbb{Z}_+$ with the lower Banach density 1.

In the study of systems which combine slow and fast motions, the averaging principle suggests that a good approximation of the slow motion on long time intervals can be obtained by averaging its parameters over the fast variables. When the slow and fast motions depend on each other (fully coupled), as is usually the case, for instance, in perturbations of Hamiltonian systems, the averaging prescription cannot always be applied, and when it does work, this is usually only in some averaged with respect to initial conditions sense. In this paper we first give necessary and sufficient conditions for the averaging principle to hold (in the above sense) and then, relying on some large deviations arguments, verify them in the case when the fast motions are hyperbolic (Axiom A) flows for each freezed slow variable. It turns out that in this situation the Lebesgue measure of initial conditions with bad averaging approximation tends to zero exponentially fast as the parameter tends to zero.

Let $\sigma$ and $\tau$ be two measure-preserving transformations of a non-atomic probability space, and Cob$(\sigma)$, Cob$(\tau)$ be the sets of their measurable coboundaries. We show that if the group G generated by $\sigma$ and $\tau$ is nilpotent and acts ergodically, then the inclusion Cob$(\sigma)\subseteq\text{Cob}(\tau)$ implies that $\sigma=\tau^n$ for some $n\in\mathbb Z$. This fact cannot be extended to solvable G. For G virtually solvable, a detailed description of the relationship between $\sigma$ and $\tau$ satisfying the inclusion Cob$(\sigma)\subseteq\text{Cob}(\tau)$ is given. In this case $\sigma$ is a generalized power of $\tau$ and is isomorphic to some $\tau^n,\ n\in\mathbb Z$.

The proofs require some study of non-free measure-preserving actions of elementary amenable groups and their stabilizers. In particular, a version of the Rokhlin lemma for non-free measure-preserving actions admitting maximal stabilizers is given.

We prove that for all rational functions f on the Riemann sphere and potential $-t\ln|f'|, t\ge 0$ all the notions of pressure introduced in Przytycki (Proc. Amer. Math. Soc.351(5) (1999), 2081–2099) coincide. In particular, we get a new simple proof of the equality between the hyperbolic Hausdorff dimension and the minimal exponent of conformal measure on a Julia set. We prove that these pressures are equal to the pressure defined with the use of periodic orbits under an assumption that there are not many periodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet–Eckmann condition. In Appendix A, we discuss the case t < 0.

For every point $\chi$ in the spectrum of the operator $(h(\delta)\xi)=\xi_{n+1}+\xi_{n-1}+\beta(\delta e^{2\pi\alpha ni}+\delta^{-1}e^{-2\pi\alpha ni})\xi_n$ on $\ell^2(\mathbb{Z})$ there exists a complex number x of modulus one such that the equation $\xi_{n+1}+\xi_{n-1}+\beta(x\delta e^{2\pi\alpha ni}+\bar{x}\delta^{-1}e^{-2\pi\alpha ni})\xi_n=\chi\xi_n$ has a non-trivial solution satisfying the condition $\limsup_{|n|\to\infty}|\xi_n|^{1/|n|}\le\delta^{-1}\beta^{-1}$ provided that $\beta,\delta>1$ and $\alpha$ satisfies the diophantine condition $\lim_{n\to\infty}|{\rm sin}\,\pi\alpha n|^{-{1/n}}=1$. The parameters $x\delta$ and $\chi$ are in the range of analytic functions which are defined on a Riemann surface covering the resolvent set of the operator h(1).

Let f and g be two commuting holomorphic self-maps of the open unit disc $\mathbb{D}$ in the complex plane with a common Wolff point $\tau\in\partial\mathbb{D}$: if these two maps agree at $\tau$ up to the third order, then $f\equiv g$.