American Institute of Mathematical Sciences

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when the rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk.

We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of Sarig [29] for surfaces. These partitions allow us to obtain symbolic coding on invariant sets of full measure for all hyperbolic measures whose Lyapunov exponents are bounded away from zero by a fixed constant. Applications include counting results for hyperbolic periodic orbits, and structure of hyperbolic measures of maximal entropy.

Let \begin{document}$(X_A,σ_{A})$\end{document} be a nontrivial irreducible shift of finite type (SFT), with \begin{document}$\mathscr{M}_A$\end{document} denoting its mapping class group: the group of flow equivalences of its mapping torus \begin{document}$\mathsf{S} X_A$\end{document}, (i.e., self homeomorphisms of \begin{document}$\mathsf{S} X_A$\end{document} which respect the direction of the suspension flow) modulo the subgroup of flow equivalences of \begin{document}$\mathsf{S} X_A$\end{document} isotopic to the identity. We develop and apply machinery (flow codes, cohomology constraints) and provide context for the study of \begin{document}$\mathscr M_A$\end{document}, and prove results including the following. \begin{document}$\mathscr{M}_A$\end{document} acts faithfully and \begin{document}$n$\end{document}-transitively (for every \begin{document}$n$\end{document} in \begin{document}$\mathbb{N}$\end{document}) by permutations on the set of circles of \begin{document}$\mathsf{S} X_A$\end{document}. The center of \begin{document}$\mathscr{M}_A$\end{document} is trivial. The outer automorphism group of \begin{document}$\mathscr{M}_A$\end{document} is nontrivial. In many cases, \begin{document}$\text{Aut}(σ_{A})$\end{document} admits a nonspatial automorphism. For every SFT \begin{document}$(X_B,σ_B)$\end{document} flow equivalent to \begin{document}$(X_A,σ_{A})$\end{document}, \begin{document}$\mathscr{M}_A$\end{document} contains embedded copies of \begin{document}${\rm{Aut}}({\sigma _B})/\left\langle {{\sigma _B}} \right\rangle $\end{document}, induced by return maps to invariant cross sections; but, elements of \begin{document}$\mathscr M_A$\end{document} not arising from flow equivalences with invariant cross sections are abundant. \begin{document}$\mathscr{M}_A$\end{document} is countable and has solvable word problem. \begin{document}$\mathscr{M}_A$\end{document} is not residually finite. Conjugacy classes of many (possibly all) involutions in \begin{document}$\mathscr M_A$\end{document} can be classified by the \begin{document}$G$\end{document}-flow equivalence classes of associated \begin{document}$G$\end{document}-SFTs, for \begin{document}$G = \mathbb{Z}/2\mathbb{Z}$\end{document}. There are many open questions.

The set of automorphisms of a one-dimensional subshift \begin{document} $(X, σ)$ \end{document} forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first examples of countable groups that cannot embed into the automorphism group of any zero entropy subshift. In particular, we show that the Baumslag-Solitar groups \begin{document} ${\rm BS}(1,n)$ \end{document} and all other groups that contain exponentially distorted elements cannot embed into \begin{document} ${\rm Aut}(X)$ \end{document} when \begin{document} $h_{{\rm top}}(X) = 0$ \end{document}. We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.

An important consequence of the theory of entropy of \begin{document}$ \mathbb{Z}$\end{document}-actions is that the events measurable with respect to the far future coincide (modulo null sets) with those measurable with respect to the distant past, and that measuring the entropy using the past will give the same value as measuring it using the future. In this paper we show that for measures invariant under multiparameter algebraic actions if the entropy attached to coarse Lyapunov foliations fail to display a stronger symmetry property of a similar type this forces the measure to be invariant under non-trivial unipotent groups. Some consequences of this phenomenon are noted.

We study the smooth self-maps \begin{document}$f$\end{document} of \begin{document}$× a$\end{document}-invariant sets \begin{document}$X\subseteq[0,1]$\end{document}. Under various assumptions we show that this forces \begin{document}$\log f'(x)/\log a∈\mathbb{Q}$\end{document} at many points in \begin{document}$X$\end{document}. Our method combines scenery flow methods and equidistribution results in the positive entropy case, where we improve previous work of the author and Shmerkin, with a new topological variant of the scenery flow which applies in the zero-entropy case.

We study hyperbolic attractors of some dynamical systems with apriori given countable Markov partitions. Assuming that contraction is stronger than expansion, we construct new Markov rectangles such that their cross-sections by unstable manifolds are Cantor sets of positive Lebesgue measure. Using new Markov partitions we develop thermodynamical formalism and prove exponential decay of correlations and related properties for certain Hölder functions. The results are based on the methods developed by Sarig [26]-[28].

We study skew products where the base is a hyperbolic automorphism of \begin{document}$\mathbb{T}^2$\end{document}, the fiber is a smooth area preserving flow on \begin{document}$\mathbb{T}^2$\end{document} with one fixed point (of high degeneracy) and the skewing function is a smooth non coboundary with non-zero integral. The fiber dynamics can be represented as a special flow over an irrational rotation and a roof function with one power singularity. We show that for a full measure set of rotations the corresponding skew product is \begin{document}$K$\end{document} and not Bernoulli. As a consequence we get a natural class of volume-preserving diffeomorphisms of \begin{document}$\mathbb{T}^4$\end{document} which are \begin{document}$K$\end{document} and not Bernoulli.

We prove that for every \begin{document}$d≠3$\end{document} there is an Anosov diffeomorphism of \begin{document}$\mathbb{T}^{d}$\end{document} which is of stable Krieger type \begin{document}${\rm III}_1$\end{document} (its Maharam extension is weakly mixing). This is done by a construction of stable type \begin{document}${\rm III}_1$\end{document} Markov measures on the golden mean shift which can be smoothly realized as a \begin{document}$C^{1}$\end{document} Anosov diffeomorphism of \begin{document}$\mathbb{T}^2$\end{document} via the construction in our earlier paper.

We show that an arbitrary factor map \begin{document}$\pi :X \to Y$\end{document} on an irreducible subshift of finite type is a composition of a finite-to-one factor code and a class degree one factor code. Using this structure theorem on infinite-to-one factor codes, we then prove that any equilibrium state \begin{document}$\nu $\end{document} on \begin{document}$Y$\end{document} for a potential function of sufficient regularity lifts to a unique measure of maximal relative entropy on \begin{document}$X$\end{document}. This answers a question raised by Boyle and Petersen (for lifts of Markov measures) and generalizes the earlier known special case of finite-to-one factor codes.