In this paper we show that the natural action of the symmetric group acting on the product space $\{0,1\}^{\mathbb{N}}$ endowed with a Bernoulli measure is approximately transitive. We also extend the result to a larger class of probability measures.

We investigate different notions of recognizability for a free monoid morphism $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{B}}^{\ast }$. Full recognizability occurs when each (aperiodic) point in ${\mathcal{B}}^{\mathbb{Z}}$ admits at most one tiling with words $\unicode[STIX]{x1D70E}(a)$, $a\in {\mathcal{A}}$. This is stronger than the classical notion of recognizability of a substitution $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{A}}^{\ast }$, where the tiling must be compatible with the language of the substitution. We show that if $|{\mathcal{A}}|=2$, or if $\unicode[STIX]{x1D70E}$’s incidence matrix has rank $|{\mathcal{A}}|$, or if $\unicode[STIX]{x1D70E}$ is permutative, then $\unicode[STIX]{x1D70E}$ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé [Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci.99(2) (1992), 327–334] and Bezuglyi et al [Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys.29(1) (2009), 37–72], by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an $S$-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the $S$-adic shift it generates, and the measurable Bratteli–Vershik dynamical system that it defines.

Gorodetski et al. [Nonremovability of zero Lyapunov exponents. Funktsional. Anal. i Prilozhen.39(1) (2005), 27–38 (in Russian); Engl. Transl. Funct. Anal. Appl.39(1) (2005), 21–30] and Bochi et al. [Robust criterion for the existence of nonhyperbolic ergodic measures. Comm. Math. Phys.344(3) (2016), 751–795] propose two very different ways for building non-hyperbolic measures, Gorodetski et al. (2005) building such a measure as the limit of periodic measures and Bochi et al. (2016) as the $\unicode[STIX]{x1D714}$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in Gorodetski et al. (2005) has been used in a generic setting in Bonatti et al. [Non-hyperbolic ergodic measures with large support. Nonlinearity23(3) (2010), 687–705] and Díaz and Gorodetski [Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes. Ergod. Th. & Dynam. Sys.29(5) (2009), 1479–1513], as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in Bochi et al. (2016) are accumulated by periodic measures. In this paper we use a shadowing lemma from Gan [A generalized shadowing lemma. Discrete Contin. Dyn. Syst.8(3) (2002), 527–632]:

• ∙ for getting the periodic orbits in Gorodetski et al. (2005) without perturbing the dynamics;

• ∙ for recovering the compact set in Bochi et al. (2016) with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits.

As a consequence, we prove that there exists an open and dense subset ${\mathcal{U}}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in {\mathcal{U}}$, there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset ${\mathcal{V}}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in {\mathcal{V}}$, there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

We study minimality for continuous actions of abelian semigroups on compact Hausdorff spaces with a free interval. First, we give a necessary and sufficient condition for such a space to admit a minimal action of a given abelian semigroup. Further, for actions of abelian semigroups we provide a trichotomy for the topological structure of minimal sets intersecting a free interval.

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to $\infty$ through a finite chain of such components.

This paper is concerned with factors of independent and identically distributed processes on the $d$-regular tree for $d\geq 3$. We study the mutual information of values on two given vertices. If the vertices are neighbors (i.e. their distance is $1$), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$, of order $(d-1)^{-k/2}$. Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process, the rate of mutual information decay is much faster, essentially of order $(d-1)^{-k}$.

Given an $\unicode[STIX]{x1D6FC}>1$ and a $\unicode[STIX]{x1D703}$ with unbounded continued fraction entries, we characterize new relations between Sturmian subshifts with slope $\unicode[STIX]{x1D703}$ with respect to (i) an $\unicode[STIX]{x1D6FC}$-Hölder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $\unicode[STIX]{x1D703}$, and (iii) complexity notions which we call $\unicode[STIX]{x1D6FC}$-repetitiveness, $\unicode[STIX]{x1D6FC}$-repulsiveness and $\unicode[STIX]{x1D6FC}$-finiteness—generalizations of the properties known as linear repetitiveness, repulsiveness and power freeness, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is $2/(\unicode[STIX]{x1D6FC}+1)$.

We construct an infinite-dimensional compact metric space $X$, which is a closed subset of $\mathbb{S}\times \mathbb{H}$, where $\mathbb{S}$ is the unit circle and $\mathbb{H}$ is the Hilbert cube, and a skew-product map $F$ acting on $X$ such that $(X,F)$ is Li–Yorke sensitive but possesses at most countable scrambled sets. This disproves the conjecture of Akin and Kolyada that Li–Yorke sensitivity implies Li–Yorke chaos [Akin and Kolyada. Li–Yorke sensitivity. Nonlinearity16, (2003), 1421–1433].

We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map $T$ from the underlying space to itself, $T$ is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of $\mathbb{Z}^{d}$ with completely positive entropy.

We establish a dynamical version of the Kuratowski–Mycielski theorem on the existence of ‘large’ invariant dependent sets. We apply this result to the study of invariant chaotic sets in topological dynamical systems, simplify many known results on this topic and also obtain some new results.

We simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets $K$ of probability measures on a Cantor space $X$ for which there exists a minimal homeomorphism of $X$ whose set of invariant measures coincides with $K$. We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz’ theorem stating that any non-empty metrizable Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of Ash.

We prove that if $n\geq 2$, then there is no $C^{1}$-diffeomorphism $f$ of the $n$-torus such that $f$ is semi-conjugate to a minimal translation and its wandering domains are geometric balls. This improves a recent result of Navas, who proved it assuming $C^{n+1}$ regularity of $f$.

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz, and R. Ures: a reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-\text{id}$, or the mapping torus of a hyperbolic automorphism.

Motivated by the classical results by Halmos and Rokhlin on the genericity of weakly but not strongly mixing transformations and the Furstenberg tower construction, we show that weakly but not strongly mixing extensions on a fixed product space with both measures non-atomic are generic. In particular, a generic extension does not have an intermediate nilfactor.