We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a map $g$ of constant slope. In particular, we show that in the case of a Markov map $f$ that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope $e^{h_{\text{top}}(f)}$, where $h_{\text{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\text{top}}(f)$ is achievable through horseshoes of the map $f$.

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

We give explicit formulae for fringe lengths of the Calegari–Walker ziggurats—i.e., graphs of extremal rotation numbers associated with positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedra on the bounded cohomology of free groups. This explains phenomena observed experimentally by Gordenko, Calegari and Walker.

We present the quadratic Weyl sums with θ,x∈[0,1) as cocycles over a measure-preserving transformation on the two-dimensional torus. We show then that these cocycles are not coboundaries for every irrational θ∈[0,1), and that for a dense Gδ set of θ∈[0,1) the corresponding skew product is ergodic. For each of those θ, there exists a dense Gδ set of full measure of x∈[0,1) for which the sequence , n=1,2,… , is dense in .

We associate to each non-singular action A of a connected Lie group G on a closed manifold M and each A-invariant probability measure \mu on M, a linear mapping P_A from the cohomology H^*(\mathcal{F}) of the foliation \mathcal{F} given by the orbits of A onto the cohomology H^*(\mathcal{G}) of the Lie algebra \mathcal{G} of G. This extends to actions the asymptotic cycles of S. Schwartzman and D. Sullivan and gives a strong connection between the ergodic properties of A and the characteristic mapping \chi of \mathcal{F} as defined by dos Santos (Contemp. Math.161 (1994), 41–57). The ergodic notion of E-action introduced by Arraut and dos Santos (Topology31 (1992), 545–555) extends naturally to actions of Lie groups and we prove the main result of this article which shows that a non-singular E-action A of a nilpotent Lie group G completely determines the characteristic mapping \chi of the foliation \mathcal{F} given by A. This shows the strength of the above connection and gives as a corollary the vanishing theorem for foliated bundles proved by dos Santos.

Let $M$ be a closed, oriented, and connected Riemannian $n$-manifold, for $n\geq 2$, which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f:M\rightarrow M$, the topological entropy $h(f)$ is $\log \deg f$. This proves Shub’s entropy conjecture in this case.

This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

In response to a question of Newhouse, we show that in many cases an Anosov automorphism can be uniformly approximated by a Smale diffeomorphism of the same topological entropy. We conjecture this can be done whenever the automorphism is sufficiently hyperbolic.

A three-dimensional example is given that suggests this approximation is not always possible.

We establish exponential decay of correlations of all orders for locally G-accessible isometric extensions of transitive Anosov flows, under the assumption that the strong stable and strong unstable distributions of the base Anosov flow are  $C^1$ . This is accomplished by translating accessibility properties of the extension into local non-integrability estimates measured by infinitesimal transitivity groups used by Dolgopyat, from which we obtain contraction properties for a class of ‘twisted’ symbolic transfer operators.

For smooth random dynamical systems we consider the quenched linear and higher-order response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gouëzel, Keller and Liverani [28, 33], which has been applied to deterministic systems with great success, may be adapted to study random systems that possess good mixing properties. As a consequence, we obtain general linear and higher-order response results, as well as the differentiability of the variance in quenched central limit theorems (CLTs), for random dynamical systems (RDSs) that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasize that our results apply to random dynamical systems over a general ergodic base map, and are obtained without resorting to infinite-dimensional multiplicative ergodic theory.

In this paper, we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same renormalized Hilbert length for two distinct convex real projective structures. The correlation number in this asymptotic formula is characterized in terms of their Manhattan curve. We show that the correlation number is not uniformly bounded away from zero on the space of pairs of hyperbolic surfaces, answering a question of Schwartz and Sharp. In contrast, we provide examples of diverging sequences, defined via cubic rays, along which the correlation number stays larger than a uniform strictly positive constant. In the last section, we extend the correlation theorem to Hitchin representations.

We introduce a new method of constructing Birkhoff sections for pseudo-Anosov flows, which uses the connection between pseudo-Anosov flows and veering triangulations. This method allows for explicit constructions, as well as control over the Birkhoff section in terms of its Euler characteristic and the complexity of the boundary orbits. In particular, we show that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation—not just a formal power series—bringing the Hamiltonian into its Birkhoff normal form.

Let f: D → D be an inner function with a fixed point p ∈ D, and f*: S1 → S1 be its extension to the unit circle. We prove in this paper that the Rohlin invertible extension of the system (f*, λp) is equivalent to a generalized Bernoulli shift, where λp is the harmonic measure associated with p.

We prove the first orbit equivalence superrigidity results for actions of type III$_\unicode{x3bb} $ when $\unicode{x3bb} \neq 1$. These actions arise as skew products of actions of dense subgroups of $\operatorname {SL}(n,\mathbb {R})$ on the sphere $S^{n-1}$ and they can have any prescribed associated flow.

All the theorems stated in [2] are true. In particular, theorem B, on the existence of an invariant measure, is true. However, in the proof of theorem B, it is necessary to use the result of G. Keller [1] on the existence of an invariant measure for a C1+α function. The attempt, in § 4 of this paper, to modify the proof of S. Wong so that it applies to this general case of a C1+α function, is faulty. These calculations do not overcome the real difficulties involved in the proof and leave major gaps in the estimates needed.

The aim of this paper is to introduce and study the class of boundedly non-additive processes. The main result is the decomposition in theorem (2.1) and theorem (3.1), which says that a boundedly non-additive process is the sum of a non-positive subadditive, a non-negative superadditive and an additive process. By this decomposition we can extend the mean and local ergodic theorems for superadditive processes of M. A. Akcoglu and U. Krengel to boundedly non-additive processes. At the end of this paper some examples are given.

We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter $\nu $: the regularized dynamics is globally defined for each $\nu> 0$, and the original singular system is recovered in the limit of vanishing $\nu $. We prove that this limit yields a unique statistical solution independent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.

Let $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$. Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$, $n\geqslant 1$. For any $t \in {\mathbb R}$, denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$. By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$. We also establish integral and local-type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

Let ℋ denote the group of homeomorphisms of a σ-compact manifold M which preserve a locally positive non-atomic measure μ. Such a manifold can be compactified by adjoining ideal points called ‘ends’, collectively denoted by E. Every homeomorphism h in ℋ induces a measure preserving system (E, , μ*, h*) where is the algebra of clopen subsets of E, μ* is a 0-∞ measure induced on E by μ, and h*: E → E is a μ*-preserving homeomorphism. For any induced homeomorphism σ = f*, where f belongs to ℋ, define ℋσ = {h ∈ ℋ: h* = σ}. We prove that ergodicity is generic in ℋσ for the compact-open topology if and only if (E, , μ*, σ) is incompressible and ergodic. Furthermore ℋσ contains an ergodic homeomorphism if and only if (E, , μ*, σ) is incompressible. Since the identity on M induces the identity on E, which is incompressible, our results establish that every manifold (M, μ) supports an ergodic μ-preserving homeomorphism.