In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

We prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu $ on M where the transition kernel ${\mathcal P}$ admits an eigenfunction $0\leq \eta \in L^1(M,\mu )$. We find conditions on the transition densities of ${\mathcal P}$ with respect to $\mu $ which ensure that $\eta (x) \mu (\mathrm {d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $\mu $-almost surely. We apply this result to the random logistic map $X_{n+1} = \omega _n X_n (1-X_n)$ absorbed at ${\mathbb R} \setminus [0,1],$ where $\omega _n$ is an independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$

We prove that if two free probability-measure-preserving (p.m.p.) ${\mathbb Z}$-actions are Shannon orbit equivalent, then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein–Weiss and the entropy invariance results of Austin and Kerr–Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also show, at the opposite end of the stochastic spectrum, that every ${\mathbb Z}$-odometer is Shannon orbit equivalent to the universal ${\mathbb Z}$-odometer.

Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim _{\mathbb {C}} M \geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal $\mathrm {U}(m)$-bundle $F_{\mathbb {C}}M$ of unitary frames. We show that if $m \geq 6$ is even and $m \neq 28$, there exists $\unicode{x3bb} (m) \in (0, 1)$ such that if $(M, g)$ has negative $\unicode{x3bb} (m)$-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $\unicode{x3bb} (m)$ satisfy $\unicode{x3bb} (6) = 0.9330...$, $\lim _{m \to +\infty } \unicode{x3bb} (m) = {11}/{12} = 0.9166...$, and $m \mapsto \unicode{x3bb} (m)$ is decreasing. This extends to the even-dimensional case the results of Brin and Gromov [On the ergodicity of frame flows. Invent. Math. 60(1) (1980), 1–7] who proved ergodicity of the unitary frame flow on negatively curved compact Kähler manifolds of odd complex dimension.

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 show how to apply classical probabilistic tools for partial sums $\sum _{j=0}^{n-1}\varphi \circ \tau ^j$ generated by a skew product $\tau $, built over a sufficiently well-mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $\varphi $, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate-deviations principle, several exponential concentration inequalities and Rosenthal-type moment estimates for skew products with $\alpha $-, $\phi $- or $\psi $-mixing base maps and expanding-on-average random fiber maps. All of the results are new even in the uniformly expanding case. The main novelty here (in contrast to [2]) is that the random maps are not independent, they do not preserve the same measure and the observable $\varphi $ depends also on the base space. For stretched exponentially ${\alpha }$-mixing base maps our proofs are based on multiple correlation estimates, which make the classical method of cumulants applicable. For $\phi $- or $\psi $-mixing base maps, we obtain an ASIP and maximal and concentration inequalities by establishing an $L^\infty $ convergence of the iterates ${\mathcal K}^{\,n}$ of a certain transfer operator ${\mathcal K}$ with respect to a certain sub-${\sigma }$-algebra, which yields an appropriate (reverse) martingale-coboundary decomposition.

In this paper, we reduce the logarithmic Sarnak conjecture to the $\{0,1\}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.

Let $\alpha $ be a $C^{\infty }$ volume-preserving action on a closed n-manifold M by a lattice $\Gamma $ in $\mathrm {SL}(n,\mathbb {R})$, $n\ge 3$. Assume that there is an element $\gamma \in \Gamma $ such that $\alpha (\gamma )$ admits a dominated splitting. We prove that the manifold M is diffeomorphic to the torus ${{\mathbb T}^{n}={\mathbb R}^{n}/{\mathbb Z}^{n}}$ and $\alpha $ is smoothly conjugate to an affine action. Anosov diffeomorphisms and partial hyperbolic diffeomorphisms admit a dominated splitting. We obtained a topological global rigidity when $\alpha $ is $C^{1}$. We also prove similar theorems for actions on $2n$-manifolds by lattices in $\textrm {Sp}(2n,{\mathbb R})$ with $n\ge 2$ and $\mathrm {SO}(n,n)$ with $n\ge 5$.

Let G be a countable residually finite group (for instance, ${\mathbb F}_2$) and let $\overleftarrow {G}$ be a totally disconnected metric compactification of G equipped with the action of G by left multiplication. For every $r\geq 1$, we construct a Toeplitz G-subshift $(X,\sigma ,G)$, which is an almost one-to-one extension of $\overleftarrow {G}$, having r ergodic measures $\nu _1, \ldots ,\nu _r$ such that for every $1\leq i\leq r$, the measure-theoretic dynamical system $(X,\sigma ,G,\nu _i)$ is isomorphic to $\overleftarrow {G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups); however, we point out the differences and obstructions that could appear when the acting group is not amenable.

We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of ${\mathcal B}$-free subshifts satisfying them, extending the work of Dymek [Automorphisms of Toeplitz ${\mathcal B}$-free systems. Bull. Pol. Acad. Sci. Math. 65(2) (2017), 139–152]. Additionally, we provide an example of a ${\mathcal B}$-free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 of S. Ferenczi et al [Sarnak’s conjecture: what’s new. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics (Lecture Notes in Mathematics, 2213). Eds. S. Ferenczi, J. Kułaga-Przymus and M. Lemańczyk. Springer, Cham, 2018, pp. 163–235].

Pavlov [Adv. Math. 295 (2016), 250–270; Nonlinearity 32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

Let $ T:[0,1]\to [0,1] $ be an expanding Markov map with a finite partition. Let $ \mu _\phi $ be the invariant Gibbs measure associated with a Hölder continuous potential $ \phi $. For $ x\in [0,1] $ and $ \kappa>0 $, we investigate the size of the uniform approximation set

$$ \begin{align*}\mathcal U^\kappa(x):=\{y\in[0,1]:\text{ for all } N\gg1, \text{ there exists } n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\}.\end{align*} $$

The critical value of $ \kappa $ such that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $-almost every (a.e.) $ x $ is proven to be $ 1/\alpha _{\max } $, where $ \alpha _{\max }=-\int \phi \,d\mu _{\max }/\int \log |T'|\,d\mu _{\max } $ and $ \mu _{\max } $ is the Gibbs measure associated with the potential $ -\log |T'| $. Moreover, when $ \kappa>1/\alpha _{\max } $, we show that for $ \mu _\phi $-a.e. $ x $, the Hausdorff dimension of $ \mathcal U^\kappa (x) $ agrees with the multifractal spectrum of $ \mu _\phi $.

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 $\Sigma $ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of probability measure preserving ergodic minimal profinite actions for the fundamental group of $\Sigma $ that are topologically free but not essentially free, a property that we call allostery. Moreover, the invariant random subgroups we obtain are pairwise distincts.

We exhibit, for arbitrary $\epsilon> 0$, subshifts admitting weakly mixing (probability) measures with word complexity p satisfying $\limsup p(q) / q < 1.5 + \epsilon $. For arbitrary $f(q) \to \infty $, said subshifts can be made to satisfy $p(q) < q + f(q)$ infinitely often. We establish that every subshift associated to a rank-one transformation (on a probability space) which is not an odometer satisfies $\limsup p(q) - 1.5q = \infty $ and that this is optimal for rank-ones.

We establish analogues for trees of results relating the density of a set ${E \subset \mathbb {N}}$, the density of its set of popular differences and the structure of E. To obtain our results, we formalize a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting, we use a recent result of the first author with Björklund and Shkredov and a stability-type extension (proved jointly with Shkredov); we also prove a new result for non-ergodic systems.

For $\mathscr {B} \subseteq \mathbb {N} $, the $ \mathscr {B} $-free subshift $ X_{\eta } $ is the orbit closure of the characteristic function of the set of $ \mathscr {B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\eta } $, have their analogues for $ X_{\eta } $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures [G. Keller. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21(3) (2021), Paper No. 2140008]. A central assumption in our work is that $\eta ^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_{\eta } $) is regular. From this, we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_{\eta } $ from above and below.

Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi $ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu $ on X and $\nu $ on Y, with $\mu $ ergodic. Let $y\in Y$ be quasi-generic for $\nu $. Then there exists a point $x\in X$ generic for $\mu $ such that the pair $(x,y)$ is quasi-generic for $\xi $. This is a generalization of a similar theorem by T. Kamae, in which $(X,T)$ and $(Y,S)$ are full shifts on finite alphabets.

We prove that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One ingredient is a result of independent interest generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism.