We construct an explicit family of finite-area, infinite-genus translation surfaces whose vertical translation flow is strongly mixing. This provides a positive answer to a question posed by Lindsey and Treviño [Discrete Contin. Dyn. Syst. 36(10) (2016), 5509–5553].

Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any q in one of these intervals, the set of points with exactly m base q expansions is non-empty and, moreover, has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli [Intersections of thick compact sets in ${\mathbb{R}}^d$. Math. Z. 301(3) (2022), 2291–2315, Theorem 6], which guarantees that the intersection of a family of compact subsets of $\mathbb {R}^d$ has positive Hausdorff dimension under certain conditions.

Given two rational maps $f,g: \mathbb {P}^1 \to \mathbb {P}^1$ of degree d over $\mathbb {C}$, DeMarco, Krieger, and Ye [Common preperiodic points for quadratic polynomials. J. Mod. Dyn. 18 (2022), 363–413] have conjectured that there should be a uniform bound $B = B(d)> 0$ such that either they have at most B common preperiodic points or they have the same set of preperiodic points. We study their conjecture from a statistical perspective and prove that the average number of shared preperiodic points is zero for monic polynomials of degree $d \geq 6$ with rational coefficients. We also investigate the quantity $\liminf _{x \in \overline {\mathbb {Q}}} (\widehat {h}_f(x) + \widehat {h}_g(x) )$ for a generic pair of polynomials and prove both lower and upper bounds for it.

We classify the sets of natural numbers n for which certain dynamical systems $(X,f)$ on a compact metric space X have a periodic point of (least) period n. Interest in this question dates back to Sharkovskii’s theorem for continuous maps on intervals of the real line, but it also ties to checkable conditions for Krieger’s embedding theorem for symbolic dynamical systems. Given a system for which the logarithmic derivative of the Artin–Mazur zeta function is rational, we use the Skolem–Mahler–Lech theorem to classify for which n the system has a periodic point of (not necessarily least) period n. Moreover, we build on work on finitely presented (FP) systems and their relationship to symbolic dynamics to classify the set of least periods, that is, periodic orbit lengths, for arbitrary FP systems, extending a known classification for shifts of finite type. We also provide several constructions to realize any such least period sets.

It is well known that the dynamical behavior of a rational map $f:\widehat {\mathbb C}\to \widehat {\mathbb C}$ is governed by the forward orbits of the critical points of f. The map f is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of f. We encode the orbits of the critical points of f with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial ramification portrait is realized as the ramification portrait of a postcritically finite polynomial and classify which abstract polynomial ramification portraits can only be realized by unobstructed maps.

The action of a finite group G on a subshift of finite type (SFT) X is called free if every point has trivial stabilizer and it is called inert if the induced action on the dimension group of X is trivial. We show that any two free inert actions of a finite group G on an SFT are conjugate by an automorphism of any sufficiently high power of the shift space. This partially answers a question posed by Fiebig. As a consequence, we obtain that every two free elements of the stabilized automorphism group of a full shift are conjugate in this group. In addition, we generalize a result of Boyle, Carlsen, and Eilers concerning the flow equivalence of G-SFTs.

Let $\Gamma $ be a torsion-free subgroup of $\mathrm {SL}_3(\mathbb {R})$ commensurable with $\mathrm {SL}_3(\mathbb {Z})$, and $Y=\mathrm {SO}_3(\mathbb {R})\backslash \mathrm {SL}_3(\mathbb {R})/\Gamma $ be endowed with the natural locally symmetric space structure. We prove that for any point $y\in Y$, the set of directions in which the geodesic ray starting from y is bounded in Y, is hyperplane absolute winning.

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and have finite co-volume. A theorem of Lagarias [Meyer’s concept of quasicrystal and quasiregular sets. Comm. Math. Phys. 179(2) (1996), 365–376] provides a criterion for discrete subsets of Euclidean spaces to be approximate lattices. It asserts that if a subset X of $\mathbb {R}^n$ is relatively dense and $X - X$ is uniformly discrete, then X is an approximate lattice. We prove two generalizations of Lagarias’ theorem: when the ambient group is amenable and when it is a higher-rank simple algebraic group over a characteristic $0$ local field. This is a natural counterpart to the recent structure results for approximate lattices in non-commutative locally compact groups. We also provide a reformulation in dynamical terms pertaining to return times of cross-sections. Our method relies on counting arguments involving the so-called periodization maps, ergodic theorems and a method of Tao regarding small doubling for finite subsets. In the case of simple algebraic groups over local fields, we moreover make use of deep superrigidity results due to Margulis and to Zimmer.

Given a Fuchsian group $\Gamma $, we introduce escape rate spectra associated with $\Gamma $ to investigate the transient behavior of the geodesic flow on the hyperbolic surface $\mathbb {D}/\Gamma $. Our definition is motivated by Bishop’s linear escape set. In the special case when $\mathbb {D}/\Gamma $ is a hyperbolic $\mathbb {Z}$-covering of a surface uniformized by a generalized Schottky group, we completely determine the escape rate spectra by the convex conjugate of a generalized Poincaré exponent.

We consider an Arnold’s cat map generated $C^1$ bounded potential with the directional derivative in the unstable direction uniformly bounded away from zero. We show that the Lyapunov exponent for the associated Schrödinger operator is uniformly positive for all energies provided the coupling is sufficiently large.

We generalize Hopf’s theorem to thermostats: the total thermostat curvature of a thermostat without conjugate points is non-positive and vanishes only if the thermostat curvature is identically zero. We further show that, if the thermostat curvature is zero, then the flow has no conjugate points and the Green bundles collapse almost everywhere. Given a thermostat without conjugate points, we prove that the Green bundles are transverse everywhere if and only if it is projectively Anosov. Finally, we provide an example showing that Hopf’s rigidity theorem on the $2$-torus cannot be extended to thermostats. It is also the first example of a projectively Anosov thermostat which is not Anosov.

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems and to some Higman–Thompson groups.

For upper semi-continuous potentials defined on shifts over countable alphabets, this paper ensures sufficient conditions for the existence of a maximizing measure. We resort to the concept of blur shift, introduced by T. Almeida and M. Sobottka as a compactification method for countable alphabet shifts consisting of adding new symbols given by blurred subsets of the alphabet. Our approach extends beyond the Markovian case to encompass more general countable alphabet shifts. In particular, we guarantee a convex characterization and compactness for the set of blur invariant probabilities with respect to the discontinuous shift map.

Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the class whose number of asymptotic components is exactly the given cardinal. For finite or countable ones, we explicitly construct such examples using $\mathcal {S}$-adic subshifts. We obtain the uncountable case by showing that any topological dynamical system with at most countably many asymptotic components has zero topological entropy. We also construct systems that have arbitrarily high subexponential word complexity, but only one asymptotic component. We deduce that within any strong orbit equivalence class, there exists a minimal subshift whose automorphism group is isomorphic to $\mathbb {Z}$.

In this paper, we consider a class of affine Anosov mappings with a quasi-periodic forcing and show that there is a unique positive integer m, which only depends on the system, such that the exponential growth rate of the number of invariant tori of degree m is equal to the topological entropy.

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu $ defined on the completion of the Borel $\sigma $-algebra, and $\mathcal F$ a f-invariant one-dimensional continuous foliation of M by $C^1$-leaves. Then, if f preserves a continuous $\mathcal {F}$-arc length system, then we only have three possibilities for the conditional measures of $\mu $ along $\mathcal F$, namely: (i) they are atomic for almost every leaf; or (ii) for almost every leaf, they are equivalent to the measure $\unicode{x3bb} _x$ induced by the invariant arc-length system over $\mathcal F$; or (iii) for almost every leaf, their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

Blenders are special hyperbolic sets used to produce various robust dynamical phenomena that appear fragile at first glance. We prove that for $C^r$ diffeomorphisms ($r=2,\ldots ,\infty ,\omega $), blenders naturally exist (without perturbation) near non-degenerate heterodimensional cycles of coindex 1, and the existence is determined by arithmetic properties of moduli of topological conjugacy for diffeomorphisms with heterodimensional cycles. In particular, we obtain a dichotomy for the dynamics in any small neighborhood U of a non-degenerate heterodimensional cycle: either there exist infinitely many blenders accumulating on the cycle, forming robust heterodimensional dynamics in most cases, or there are no orbits, other than those constituting the cycle, lying entirely in U.

We consider a DA-type surgery of the famous Lorenz attractor in dimension 4. This kind of surgery was first used by Smale [Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73(6) (1967), 747–817] and Mañé [Contributions to the stability conjecture. Topology 17(4) (1978), 383–396] to give important examples in the study of partially hyperbolic systems. Our construction gives the first example of a singular chain recurrence class which is Lyapunov stable, away from homoclinic tangencies, and exhibits robustly heterodimensional cycles. Moreover, the chain recurrence class has the following interesting property: there exists robustly a two-dimensional sectionally expanding subbundle (containing the flow direction) of the tangent bundle such that it is properly included in a subbundle of the finest dominated splitting for the tangent flow.

A power-bounded operator T satisfying $\sup _n n\lVert T^n-T^{n+1}\rVert <\infty $ is a Ritt operator. For such operators, we study the generalized square function $Q_{\alpha ,s,r}^Tf=(\sum _n n^{\alpha } |T^n(I-T)^rf|^s)^{1/s}$. It is known that when T is a positive contraction and a Ritt operator on $L^p$, $1<p<\infty $, then for any integer $r\ge 1$, the square function $Q_{2r-1,2,r}^Tf$ defines a bounded operator [17] on $L^p$. In this work, we extend the theory to the endpoint case $p=1$. We show that if T is a Ritt operator on $L^1$, then the generalized square function $Q_{\alpha ,s,r}^Tf $ is bounded on $L^1$ whenever $\alpha +1<sr$. In the particular setting where T is a convolution operator of the form $T_{\mu }=\sum _k \mu (k) U^kf$, with $\mu $ a probability measure on $\mathbb Z$ and U the composition operator induced by an invertible measure-preserving transformation, we provide sufficient conditions on $\mu $ under which $Q_{2r-1,2,r}^{T_{\mu }}f$ is of weak type $(1,1)$, for $r>0$. We also establish bounds for variational and oscillation norms, $\lVert n^{\beta } T^n(1-T)^r\rVert _{v(s)}$ and $\lVert n^{\beta } T^n(1-T)^r\rVert _{o(s)}$, for Ritt operators, highlighting endpoint behavior.