We study the billiard dynamics in annular tables between two eccentric circles. As the center and the radius of the inner circle changes, a two-parameters map is defined by the first return of trajectories to the obstacle. We obtain an increasing family of hyperbolic sets, in the sense of the Hausdorff distance, as the radius goes to zero and the center of the obstacle approximates the outer boundary. The dynamics on each of these sets is conjugate to a shift with an increasing number of symbols. We also show that for many parameters, the system presents quadratic homoclinic tangencies whose bifurcation gives rise to elliptical islands (conservative Newhouse phenomenon). Thus, for many parameters, we obtain the coexistence of a ‘large’ hyperbolic set with many elliptical islands.

We investigate several questions related to the notion of recognizable morphism. The main result is a new proof of Mossé’s theorem and actually of a generalization to a more general class of morphisms due to Berthé et al [Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys. 39(11) (2019), 2896–2931]. We actually prove the result of Berthé et al for the most general class of morphisms, including ones with erasable letters. Our result is derived from a result concerning elementary morphisms for which we also provide a new proof. We also prove some new results which allow us to formulate the property of recognizability in terms of finite automata. We use this characterization to show that for an injective morphism, the property of being recognizable on the full shift for aperiodic points is decidable.

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.

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

We study the asymptotic behavior of the sequence $ \{\Omega (n) \}_{ n \in \mathbb {N} } $ from a dynamical point of view, where $ \Omega (n) $ denotes the number of prime factors of $ n $ counted with multiplicity. First, we show that for any non-atomic ergodic system $(X, \mathcal {B}, \mu , T)$, the operators $T^{\Omega (n)}: \mathcal {B} \to L^1(\mu )$ have the strong sweeping-out property. In particular, this implies that the pointwise ergodic theorem does not hold along $\Omega (n)$. Second, we show that the behaviors of $\Omega (n)$ captured by the prime number theorem and Erdős–Kac theorem are disjoint, in the sense that their dynamical correlations tend to zero.

We give an example of an FIID vertex-labeling of ${\mathbb T}_3$ whose marginals are uniform on $[0,1]$, and if we delete the edges between those vertices whose labels are different, then some of the remaining clusters are infinite. We also show that no such process can be finitary.

In this paper, we address the problem of computing the topological entropy of a map $\psi : G \to G$, where G is a Lie group, given by some power $\psi (g) = g^k$, with k a positive integer. When G is abelian, $\psi $ is an endomorphism and its topological entropy is given by $h(\psi ) = \dim (T(G)) \log (k)$, where $T(G)$ is the maximal torus of G, as shown by Patrão [The topological entropy of endomorphisms of Lie groups. Israel J. Math. 234 (2019), 55–80]. However, when G is not abelian, $\psi $ is no longer an endomorphism and these previous results cannot be used. Still, $\psi $ has some interesting symmetries, for example, it commutes with the conjugations of G. In this paper, the structure theory of Lie groups is used to show that $h(\psi ) = \dim (T)\log (k)$, where T is a maximal torus of G, generalizing the formula in the abelian case. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.

We consider the McMullen maps $f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$ with $\unicode{x3bb} \in \mathbb {C}^{*}$ and $n \geq 3$. We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and Sierpiński holes) are quasi-circles with Hausdorff dimension strictly between $1$ and $2$.

We consider the family $\mathrm {MC}_d$ of monic centered polynomials of one complex variable with degree $d \geq 2$, and study the map $\widehat {\Phi }_d:\mathrm {MC}_d\to \widetilde {\Lambda }_d \subset \mathbb {C}^d / \mathfrak {S}_d$ which maps each $f \in \mathrm {MC}_d$ to its unordered collection of fixed-point multipliers. We give an explicit formula for counting the number of elements of each fiber $\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$ for every $\bar {\unicode{x3bb} } \in \widetilde {\Lambda }_d$ except when the fiber $\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$ contains polynomials having multiple fixed points. This formula is not a recursive one, and is a drastic improvement of our previous result [T. Sugiyama. The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math. 322 (2017), 132–185] which gave a rather long algorithm with some induction processes.

Let $u_{X}^{t}$ be a unipotent flow on $X=\mathrm {SO}(n,1)/\Gamma $, $u_{Y}^{t}$ be a unipotent flow on $Y=G/\Gamma ^{\prime }$. Let $\tilde {u}_{X}^{t}$, $\tilde {u}_{Y}^{t}$ be time changes of $u_{X}^{t}$, $u_{Y}^{t}$, respectively. We show the disjointness (in the sense of Furstenberg) between $u_{X}^{t}$ and $\tilde {u}_{Y}^{t}$ (or $\tilde {u}_{X}^{t}$ and $u_{Y}^{t}$) in certain situations. Our method refines the works of Ratner’s shearing argument. The method also extends a recent work of Dong, Kanigowski, and Wei [Rigidity of joinings for some measure preserving systems. Ergod. Th. & Dynam. Sys. 42 (2022), 665–690].

We construct a geometrico-symbolic version of the natural extension of the random $\beta $-transformation introduced by Dajani and Kraaikamp [Random $\beta $-expansions. Ergod. Th. & Dynam. Sys. 23(2) (2003) 461–479]. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random $\beta $-transformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky [$\beta $-automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar. 24 (1973), 273–278] about the greedy transformation.