It has been conjectured that a generic diffeomorphism on a compact manifold will have trivial centralizer. In this paper we show that the conjecture is true within certain classes of partially hyperbolic diffeomorphisms.

We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is expressed by a geometric condition on the cocycle's behaviour over periodic points and associated homoclinic orbits. It allows us to prove that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Hölder continuous potentials), all Oseledets subspaces are one-dimensional. Moreover, the complement of this subset has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters.

This improves previous results of Bonatti, Gomez–Mont and Viana where it was shown that some Lyapunov exponent is non-zero, in a similar setting and also for an open dense subset.

A smooth $\mathbb{Z}^p$-action on a closed orientable manifold M is cohomologically rigid (CR) if the cohomology of the induced $\mathbb{Z}^p$-module on the smooth functions on M is isomorphic to the real cohomology of the torus Tp. We prove some general properties of a smooth $\mathbb{Z}^p$-action whose first cohomology is isomorphic to Rp. When the manifold is a low-dimensional torus Tq, $1 \le q \le 2$, we prove that any minimal (all orbits are dense) smooth $\mathbb{Z}^p$-action on Tq whose first cohomology is isomorphic to Rp is $C^\infty$ conjugate to an affine $\mathbb{Z}^p$-action. As a corollary we show that the CR $\mathbb{Z}^p$-actions on Tq, $1 \le q \le 2$, are smooth conjugations of affine CR actions.

We classify the $C^\infty$ volume-preserving uniformly quasiconformal Anosov flows, such that $E^+\oplus E^-$ is $C^\infty$ and the dimensions of E+ and E- are at least two. Then we deduce a classification of volume-preserving uniformly quasiconformal Anosov flows with smooth distributions.

We consider piecewise monotone maps of the interval with zero entropy or no periodic points. First, we give a rigid model for these maps: the interval translations mappings, possibly with flips. It follows, for example, that the complexity of a piecewise monotone map of the interval is at most polynomial if and only if this map has a finite number of periodic points up to monotone equivalence. Second, we study the invariant and ergodic measures of a piecewise monotone map with zero entropy and prove that their number is bounded by twice the number of monotony intervals; for a piecewise increasing map their number is at most the number of intervals.

This is a partial survey of various developments of the classical KAM theory in dynamical systems to infinite-dimensional lattice models and Hamiltonian PDEs. Particular emphasis is given on Nekhoroshev stability and growth of higher Sobolev norms. Some new results are also presented.

If $\mathbb{M}$ is a monoid and $\mathcal{A}$ is an Abelian group, then $\mathcal{A}^{\mathbb{M}}$ is a compact Abelian group; a linear cellular automaton (LCA) is a continuous endomorphism $\mathfrak{F}:\mathcal{A}^{\mathbb{M}}\longrightarrow \mathcal{A}^{\mathbb{M}}$ that commutes with all shift maps. If $\mathfrak{F}$ is diffusive, and $\mu$ is a harmonically mixing (HM) probability measure on $\mathcal{A}^{\mathbb{M}}$, then the sequence $\{\mathfrak{F}^N \mu\}_{N=1}^{\infty}$ weak* converges to the Haar measure on $\mathcal{A}^{\mathbb{M}}$ in density. Fully supported Markov measures on $\mathcal{A}^{\mathbb{Z}}$ are HM and non-trivial LCA on $\mathcal{A}^{(\mathbb{Z}^D)}$ are diffusive when $\mathcal{A}=\mathbb{Z}_{/p}$ is a prime cyclic group.

In the present work, we provide sufficient conditions for diffusion of LCA on $\mathcal{A}^{(\mathbb{Z}^D)}$ when $\mathcal{A}=\mathbb{Z}_{/n}$ is any cyclic group or when $\mathcal{A}=(\mathbb{Z}_{/p^r})^J$ (p prime). We also show that any fully supported Markov random field on $\mathcal{A}^{(\mathbb{Z}^D)}$ is HM (where $\mathcal{A}$ is any Abelian group).

In this paper we show that any surface in $\mathbb{R}^3$ can be modified by gluing on small ‘focusing caps’ so that its geodesic flow becomes ergodic. A new concept, finite horizon cap geometry, is what makes the construction work.

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 introduce a non-commutative generalization of the notion of (approximately proper) equivalence relations and propose the construction of a ‘quotient space’. We then consider certain one-parameter groups of automorphisms of the resulting C*-algebra and prove the existence of KMS states at every temperature. In a model originating from thermodynamics we prove that these states are unique as well. We also show a relationship between maximizing measures (the analogue of the Aubry–Mather measures for expanding maps) and ground states. In the last section we explore an interesting example of phase transitions.

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.

It is proved that special flows over irrational rotations and under functions whose Fourier coefficients are of order O(1/|n|) are disjoint in the sense of Furstenberg from all mixing flows. This is an essential strengthening of a classical result by Kočergin on the absence of mixing of special flows built over irrational rotations and under bounded variation roof functions.

We prove that a rank-one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences, in the sense that the mean ergodic theorem holds for a family of what we call dynamical sequences. The application of our theorem shows that the class of polynomial rank-one transformations, rank-one transformations where the spacers are chosen to be the values of a polynomial with some mild conditions on the polynomials, that have restricted growth are mixing transformations, implying, in particular, Adams' result on staircase transformations. Another application yields a new proof that Ornstein's class of rank-one transformations constructed using ‘random spacers’ are almost surely mixing transformations.

The dynamics of a system of two diffusively coupled identical unimodal maps is investigated. The unimodal maps have simple kneading sequences which imply that their orbits are asymptotically periodic with period a power of two. Analogous results are obtained for the coupled system for all coupling parameters. We prove the eventual periodicity of the position of orbits with respect to the diagonal of the square phase space and the asymptotic periodicity for orbits whose coordinates have the same sign. Finally, we give a global condition for the existence of symmetric orbits that are examples of orbits with this asymptotic property.

Consider a closed, smooth manifold M of non-positive curvature. Write $p:\textit{UM}\rightarrow M$ for the unit tangent bundle over M and let ${\mathcal R}_>$ denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow $\phi$ on UM. We will define the structured dimension s-dim $\mathcal{R}_>$ which, essentially, is the dimension of the set $p(\mathcal{R}_>)$ of base points of $\mathcal{R}_>$.

The main result of this paper holds for manifolds with s-dim $\mathcal{R}_><\dim M/2$: for every $\epsilon>0$, there is an $\epsilon$-dense, flow invariant, closed subset $\Xi_\epsilon\subset \textit{UM}\backslash{\mathcal{R}}_>$ such that $p(\Xi_\epsilon)=M$.

We establish rather general and simple theorems implying, among other things, the pseudo-(un)stable manifold theorem, Sternberg's theorem on smooth conjugacy between hyperbolic germs of maps or vector fields, and results of Fenichel, Hirsch, Pugh and Shub on existence, uniqueness and structural stability of stable or unstable manifolds at compact invariant manifolds. The Perron–Irwin approach via sequence spaces plays a crucial role.

Let $\Gamma$ be a lattice in SU(n, 1). For each loxodromic element $\gamma_0\in\Gamma$ we define a closed curve $\{\gamma_0\}$ on $\Gamma\backslash SU(n,1)$ that projects to the closed geodesic on the factor of the complex hyperbolic space $\Gamma\backslash{\mathbb H}^n_{{\mathbb C}}$ associated with $\gamma_0$. We prove that the cohomological equation $\mathfrak{D} F=f$ has a solution if f is the lift of a holomorphic cusp form to SU(n, 1) under the following condition: for each restriction of f to $\{\gamma_0\}$ a finite number of Fourier coefficients vanish, and this finite number grows linearly with the length of the curve. This is a generalization of the classical Livshitz theorem for SU(1, 1) (A. Livshitz. Mat. Zametki10 (1971), 555–564) where the curves are the closed geodesics themselves and the vanishing of the integrals of f over them, i.e. the zeroth Fourier coefficients, is both necessary and sufficient. An application of our result to the construction of spanning sets for spaces of holomorphic cusp forms on complex hyperbolic spaces is given in Appendix A.

We prove several new results about AF-equivalence relations and relate these to Cantor minimal systems (i.e. to minimal Z-actions). The results we obtain turn out to be crucial for the study of the topological orbit structure of more general countable group actions (as homeomorphisms) on Cantor sets, which will be the topic of a forthcoming paper. In all this, Bratteli diagrams and their dynamical interpretation are indispensable tools.

For each of the following properties, there is an isometric generalized interval exchange transformation (i.e. isometric GIET) having such property: (a) non-trivial recurrence orbits are exceptional and the union of them is a dense set, moreover the intersection of the closure of two such orbits is the union of finite orbits; (b) coexistence of dense orbits and exceptional orbits; (c) existence of a dense sequence of exceptional orbits $\{\mathcal{O}(p_k)\colon k=1,2,\dotsc\}$ such that $\overline{\mathcal{O}(p_1)}\subsetneqq\overline{\mathcal{O}(p_2)}\subsetneqq\dotsb\subsetneqq\overline{\mathcal{O}(p_k)}\subsetneqq\dotsb$.

Moreover, the isometric GIET can be suspended to a smooth foliation, without singularities, on a 2-manifold. The exceptional (respectively dense) orbits of the GIET give rise to exceptional (respectively dense) leaves of the foliation. Finite genus 2-manifolds cannot support orientable foliations with the considered dynamics.

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.