We find a countable partition P on a Lebesgue space, labeled {1,2,3,…}, for any non-periodic measure-preserving transformation T such that P generates T and, for the T,P process, if you see an n on time −1 then you only have to look at times −n,1−n,…−1 to know the positive integer i to put at time 0 . We alter that proof to extend every non-periodic T to a uniform martingale (i.e. continuous g function) on an infinite alphabet. If T has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an independent identically distributed process on ℤ2 when you read from right to left but where each column determines the next if you read left to right.

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.

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.

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.

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 this paper we prove bounds for ergodic averages for nilflows on general higher-step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become equidistributed at polynomial speed. We analyze the rate of decay which is determined by the number of steps and structure of general nilpotent Lie algebras. Our main result follows from the technique of controlling scaling operators in irreducible representations and measure estimation on close return orbits on general nilmanifolds.

Let δ be a closed *-derivation on a commutative C*-algebra , suppose that is dense in for some n = 1, 2, …, ∞, and let be a linear operator satisfying the locality condition

It is shown that , for some finite integer p≤n and functions lm on X. Estimates on the coefficients lm are obtained and applied to flows and local flows.

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.

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.

The manifold M is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$, and f is a $C^{1+\beta }$ ($\kern0.3pt\beta>0$) diffeomorphism of M. $\varphi $ is a Hölder continuous potential on M. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $\varphi $), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class $H_\chi (p)$, we prove that $\varphi $ admits a local equilibrium state on $H_\chi (p)$ if and only if $\varphi $ is ‘recurrent on $H_\chi (p)$’ (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitute as its conditional measures. An immediate corollary is the local product structure of hyperbolic equilibrium states. Second, we prove a Ledrappier–Young property for hyperbolic equilibrium states: if $\varphi $ admits a conformal family of leaf measures and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to $\varphi $) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic Sinai–Ruelle–Bowen measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure zero) has conditional measures on local unstable leaves which are absolutely continuous with respect to the Riemannian volume of these leaves.

For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

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 $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

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.

The construction of a spectral cocycle from the case of one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A spectral cocycle for substitution systems and translation flows. J. Anal. Math. 141(1) (2020), 165–205] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$, allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following the work of Treviño [Quantitative weak mixing for random substitution tilings. Israel J. Math., to appear], in the simpler, non-random setting. We review some of the results of Treviño in this special case and illustrate them on concrete examples.

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.

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.

We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curves, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, and commuting holomorphic and antiholomorphic parabolic germs.

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.

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.