Stable accessibility of partially hyperbolic systems is central to their stable ergodicity, and we establish its $C^1$ -density among partially hyperbolic flows, as well as in the categories of volume-preserving, symplectic, and contact partially hyperbolic flows. As applications, we obtain on one hand in each of these four categories of flows the $C^1$ -density of the $C^1$ -stable topological transitivity

A holomorphic endomorphism of ${{\mathbb {CP}}}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map along a periodic cycle. When $n=1$, a well-known fact is that the eigenvalue along a periodic cycle of a

For a continuous function $f:[0,1] \to [0,1]$ we define a splitting sequence admitted by f and show that the inverse limit of f is an arc if and only if f does not admit a splitting sequence.

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

We give a necessary and sufficient condition on $\beta $ of the natural extension of a $\beta $-shift, so that any equilibrium measure for a function of bounded total oscillations is a weak Gibbs measure.

Let M be a geometrically finite acylindrical hyperbolic $3$-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209 (2017), 425–461;

We show that $C^r $ generically in the space of $C^r$ conservative diffeomorphisms of a compact surface, every hyperbolic periodic point has a transverse homoclinic orbit.

We construct a sofic approximation of ${\mathbb F}_2\times {\mathbb F}_2$ that is not essentially a ‘branched cover’ of a sofic approximation by homomorphisms. This answers a question of L. Bowen.

We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $\mathbb {R}$-actions, coming from a single jointly measurable measure-preserving $\mathbb {R}^2$-action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert

We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$. Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

We introduce two properties: strong R-property and $C(q)$-property, describing a special way of divergence of nearby trajectories for an abstract measure-preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$-property. Moreover, we show that if $u_t$ is a unipotent flow on $G/\Gamma $ with $\Gamma $ irreducible

We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down

Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy

We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was studied by Choquet and Deny [Sur l’équation de convolution $\mu = \mu * \sigma $. C. R. Acad. Sci. Paris 250 (1960), 799–801] in the context of random walks generated

We prove that if a closed hyperbolic $3$-manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.

We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda

In this paper we investigate the Margulis–Ruelle inequality for general Riemannian manifolds (possibly non-compact and with a boundary) and show that it always holds under an integrable condition.

Given a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$, we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$-sided fundamental polygon of the surface S and its genus. Using this, we analyze how the

We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system $(X, \mu , T)$ with a compatible metric d. We prove that under some regularity conditions, the $\mu $-measure of the following set $$\begin{align*}R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\mathbb{N} \} \end{align*}$$ obeys a zero–full law according to the convergence

We define a family $\mathcal {B}(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set $\mathcal {B}(t)$ changes as the parameter t ranges in $[0,1]$, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have a Cantor spectrum for a generic continuous sampling map.

We construct an example of a Hamiltonian flow $f^t$ on a four-dimensional smooth manifold $\mathcal {M}$ which after being restricted to an energy surface $\mathcal {M}_e$ demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense $f^t$-invariant subset $U\subset \mathcal {M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions

We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense

We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps.

This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant

We characterize dendrites D such that a continuous selfmap of D is generically chaotic (in the sense of Lasota) if and only if it is generically ${\varepsilon }$-chaotic for some ${\varepsilon }>0$. In other words, we characterize dendrites on which generic chaos of a continuous map can be described in terms of the behaviour of subdendrites with non-empty interiors under iterates of the map. A dendrite

A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.

Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is

We use Lerman’s contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed $3$-manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the $2$-disc constructed by Fayad and Katok embed into the Reeb flow of a dynamically convex contact form on

We show that the ergodic integrals for the horocycle flow on the two-torus associated by Giulietti and Liverani with an Anosov diffeomorphism either grow linearly or are bounded; in other words, there are no deviations. For this, we use the topological invariance of the Artin–Mazur zeta function to exclude resonances outside the open unit disc. Transfer operators acting on suitable spaces of anisotropic

We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha \colon G \to {\text{Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let $A^{\alpha}$ be the fixed point algebra. Then the radius of comparison satisfies ${\text{rc}} (A^{\alpha }) \leq {\text{rc}} (A)$ and ${\text{rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{\text{card}

It is shown that for a dense $G_\delta $-subset of the subgroup of non-singular transformations (of a standard infinite $\sigma $-finite measure space) whose Poisson suspensions are non-singular, the corresponding Poisson suspensions are ergodic and of Krieger’s type III1.

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also

In this paper we study a Fermi–Ulam model where a pingpong ball bounces elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion $f(t)$ is 1-periodic and piecewise $C^3$ with a singularity, $\dot {f}(0+)\ne \dot {f}(1-)$. If the second derivative $\ddot {f}(t)$ of the platform motion is either always positive or always less than $-g$, where g is

For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$, the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected

It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of non-singular infinite direct products T of transformations$T_n$,$n\in \mathbb N$, of finite type is studied. It is shown that if$T_n$ is mildly mixing,$n\in \mathbb N$, the sequence of Radon–Nikodym derivatives of$T_n$ is asymptotically translation

Let $\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function $F:X\to \mathbb {R}$ there is a one-parameter group $\alpha ^{F}$ of automorphisms (or a flow) on the crossed product $C^*$-algebra $C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and U is the canonical unitary in the construction of the crossed product

We prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.

Let f be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu $. We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of $\mu $ in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also discuss extensions to the $C^{1+\alpha

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits

In this paper we investigate the range of validity of Ruelle’s property. First, we show that every finitely generated Fuchsian group has Ruelle’s property. We also prove the existence of an infinitely generated Fuchsian group satisfying Ruelle’s property. Concerning the negative results, we first generalize Astala and Zinsmeister’s results [Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci

We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of

We consider continuous free semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$, subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$, where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating

We study Kakutani equivalence for products of some special flows over rotations with roof function smooth except a singularity at $0\in \mathbb {T}$. We estimate the Kakutani invariant for products of these flows with different powers of singularities and rotations from a full measure set. As a corollary, we obtain a countable family of pairwise non-Kakutani equivalent products of special flows over

We prove that, if f is a homeomorphism of the 2-torus isotopic to the identity whose rotation set is a non-degenerate segment and f has a periodic point, then it has uniformly bounded deviations in the direction perpendicular to the segment.

In a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys, to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve

In this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$, where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts

For every $m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $. We also establish non-archimedean and

Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$. In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological

For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta $-transformations and random Lüroth maps with

Given a G-flow X, let $\mathrm{Aut}(G, X)$, or simply $\mathrm{Aut}(X)$, denote the group of homeomorphisms of X which commute with the G action. We show that for any pair of countable groups G and H with G infinite, there is a minimal, free, Cantor G-flow X so that H embeds into $\mathrm{Aut}(X)$. This generalizes results of [2, 7].

If $\mathcal {A}$ is a finite set (alphabet), the shift dynamical system consists of the space $\mathcal {A}^{\mathbb {N}}$ of sequences with entries in $\mathcal {A}$, along with the left shift operator S. Closed S-invariant subsets are called subshifts and arise naturally as encodings of other systems. In this paper, we study the number of ergodic measures for transitive subshifts under a condition

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups

Bayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys. 35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358 (2006), 5083–5117], who constructed a non-chaotic frequently

We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$. We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$, we show that it has a continuous invariant family of $\gamma $-Hölder Riemannian metrics on $\mathcal {M}$. We establish continuity of a measurable