Let f: D → D be an inner function with a fixed point p ∈ D, and f*: S1 → S1 be its extension to the unit circle. We prove in this paper that the Rohlin invertible extension of the system (f*, λp) is equivalent to a generalized Bernoulli shift, where λp is the harmonic measure associated with p.

In this paper, we discuss a connection between geometric measure theory and number theory. This method brings a new point of view for some number-theoretic problems concerning digit expansions. Among other results, we show that for each integer k, there is a number $M>0$ such that if $b_{1},\ldots ,b_{k}$ are multiplicatively independent integers greater than M, there are infinitely many integers whose base $b_{1},b_{2},\ldots ,b_{k}$ expansions all do not have any zero digits.

Given $\beta \in (1,2]$ , let $T_{\beta }$ be the $\beta $ -transformation on the unit circle $[0,1)$ such that $T_{\beta }(x)=\beta x\pmod 1$ . For each $t\in [0,1)$ , let $K_{\beta }(t)$ be the survivor set consisting of all $x\in [0,1)$ whose orbit $\{T^{n}_{\beta }(x): n\ge 0\}$ never hits the open interval $(0,t)$ . Kalle et al [Ergod. Th. & Dynam. Sys. 40(9) (2020) 2482–2514] proved that the Hausdorff dimension function $t\mapsto \dim _{H} K_{\beta }(t)$ is a non-increasing Devil’s staircase. So there exists a critical value $\tau (\beta )$ such that $\dim _{H} K_{\beta }(t)>0$ if and only if $t<\tau (\beta )$ . In this paper, we determine the critical value $\tau (\beta )$ for all $\beta \in (1,2]$ , answering a question of Kalle et al (2020). For example, we find that for the Komornik–Loreti constant $\beta \approx 1.78723$ , we have $\tau (\beta )=(2-\beta )/(\beta -1)$ . Furthermore, we show that (i) the function $\tau : \beta \mapsto \tau (\beta )$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau $ has no downward jumps, with $\tau (1+)=0$ and $\tau (2)=1/2$ ; and (iii) there exists an open set $O\subset (1,2]$ , whose complement $(1,2]\setminus O$ has zero Hausdorff dimension, such that $\tau $ is real-analytic, convex, and strictly decreasing on each connected component of O. Consequently, the dimension $\dim _{H} K_{\beta }(t)$ is not jointly continuous in $\beta $ and t. Our strategy to find the critical value $\tau (\beta )$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems.

We formulate and prove a precise statement of asymptotic shrinking of ‘carrot-fields’ around the Mandelbrot set M. This phenomenon was suggested in a seminal text on polynomial-like mappings by Douady and Hubbard [On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. t.18 (1985), 287–343]. This is helpful for understanding how copies of M sit in the bifurcation loci of families of rational maps.

The aim of this paper is to show $J$-stability of expanding rational maps over an algebraically closed, complete and non-Archimedean field of characteristic zero. More precisely, we will show that for any expanding rational map, there exists a neighborhood of it such that the dynamics on the Julia set of any rational map in the neighborhood is the same as the dynamics of the expanding rational map as a non-Archimedean analogue of a corollary of Mañé, Sad and Sullivan’s result [On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4)16 (1983), 193–217] in complex dynamics.

We produce infinitely many examples of Anosov flows in closed $3$-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one other closed orbit. The examples are obtained by Dehn surgery on geodesic flows. The manifolds are toroidal and have Seifert pieces and atoroidal pieces in their torus decompositions.

In a recent article, Manouchehri proved a ‘Sternberg theorem’ for Liouville vector fields and noticed that it provided normal forms for implicit differential equations and first-order partial differential equations. We establish local and global versions of Moser's celebrated result on volume and symplectic forms when they admit a non-trivial one parameter (pseudo-) group of homotheties—by definition, a Liouville field is the generator of such a flow. The local version implies that two germs of Liouville fields of a symplectic or volume form are conjugate by a diffeomorphism germ which preserves the form if and only if they are conjugate. This contains Manouchehri's theorem.

We present the quadratic Weyl sums with θ,x∈[0,1) as cocycles over a measure-preserving transformation on the two-dimensional torus. We show then that these cocycles are not coboundaries for every irrational θ∈[0,1), and that for a dense Gδ set of θ∈[0,1) the corresponding skew product is ergodic. For each of those θ, there exists a dense Gδ set of full measure of x∈[0,1) for which the sequence , n=1,2,… , is dense in .

The enumeration of combinatorial classes of the complex polynomial vector fields in $ \mathbb{C} $ presented by K. Dias [Enumerating combinatorial classes of the complex polynomial vector fields in $ \mathbb{C} $. Ergod. Th. & Dynam. Sys. 33 (2013), 416–440] is extended here to a closed form enumeration of combinatorial classes for degree $d$ polynomial vector fields up to rotations of the $2(d- 1)\mathrm{th} $ roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V. A. Liskovets [Reductive enumeration under mutually orthogonal group actions. Acta Appl. Math. 52 (1998), 91–120].

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

We consider dynamical systems, consisting of $\mathbb{Z}^{2}$-actions by continuous automorphisms on shift-invariant subgroups of $\mathbb{F}_{p}^{\mathbb{Z}^{2}}$, where $\mathbb{F}_{p}$ is the field of order $p$. These systems provide natural generalizations of Ledrappier’s system, which was the first example of a 2-mixing $\mathbb{Z}^{2}$-action that is not 3-mixing. Extending the results from our previous work on Ledrappier’s example, we show that, under quite mild conditions (namely, 2-mixing and that the subgroup defining the system is a principal Markov subgroup), these systems are almost strongly mixing of every order in the following sense: for each order, one just needs to avoid certain effectively computable logarithmically small sets of times at which there is a substantial deviation from mixing of this order.

Let ${\mathcal{R}}$ be a strongly compact $C^{2}$ map defined in an open subset of an infinite-dimensional Banach space such that the image of its derivative $D_{F}{\mathcal{R}}$ is dense for every $F$. Let $\unicode[STIX]{x1D6FA}$ be a compact, forward invariant and partially hyperbolic set of ${\mathcal{R}}$ such that${\mathcal{R}}:\unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6FA}$ is onto. The $\unicode[STIX]{x1D6FF}$-shadow $W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$ of $\unicode[STIX]{x1D6FA}$ is the union of the sets

$$\begin{eqnarray}W_{\unicode[STIX]{x1D6FF}}^{s}(G)=\{F:\operatorname{dist}({\mathcal{R}}^{i}F,{\mathcal{R}}^{i}G)\leq \unicode[STIX]{x1D6FF}\text{for every }i\geq 0\},\end{eqnarray}$$

$G\in \unicode[STIX]{x1D6FA}$

$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$

$C^{1+\text{Lip}}$

$n$

$M$

$\unicode[STIX]{x1D6FA}$

$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$

$M\cap W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$

$M$

$n$

$\unicode[STIX]{x1D6FF}^{\prime }$

$$\begin{eqnarray}\mathop{\bigcup }_{i\geq 0}{\mathcal{R}}^{-i}W_{\unicode[STIX]{x1D6FF}^{\prime }}^{s}(\unicode[STIX]{x1D6FA})\end{eqnarray}$$

$C^{k}$

$k\geq 0$

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a map $g$ of constant slope. In particular, we show that in the case of a Markov map $f$ that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope $e^{h_{\text{top}}(f)}$, where $h_{\text{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\text{top}}(f)$ is achievable through horseshoes of the map $f$.

Recently, Naito considered quasi-periodic orbits of Hölder continuous functions and obtained results relating the exponent in the Hölder condition to the asymptotic behaviour of the inclusion lengths of the $\epsilon$-almost periods of these orbits, and also to the fractal dimension of these orbits. In this paper we improve these results.

We consider homeomorphisms $f,h$ generating a faithful $\mathit{BS}(1,n)$-action on a closed surface $S$, that is, $hfh^{-1}=f^{n}$ for some $n\geq 2$. According to Guelman and Liousse [Actions of Baumslag–Solitar groups on surfaces. Discrete Contin. Dyn. Syst. A 5 (2013), 1945–1964], after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\unicode[STIX]{x1D6EC}$ of the action, included in $\text{Fix}(f)$. Here, we suppose that $f$ and $h$ are $C^{1}$ in a neighborhood of $\unicode[STIX]{x1D6EC}$ and any point $x\in \unicode[STIX]{x1D6EC}$ admits an $h$-unstable manifold $W^{u}(x)$. Using Bonatti’s techniques, we prove that either there exists an integer $N$ such that $W^{u}(x)$ is included in $\text{Fix}(f^{N})$ or there is a lower bound for the norm of the differential of $h$ depending only on $n$ and the Riemannian metric on $S$. Combining the last statement with a result of Alonso, Guelman and Xavier [Actions of solvable Baumslag–Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete Contin. Dyn. Syst. to appear], we show that any faithful action of $\mathit{BS}(1,n)$ on $S$ with $h$ a pseudo-Anosov homeomorphism has a finite orbit containing singularities of $h$; moreover, if $f$ is isotopic to the identity, it is entirely contained in the singular set of $h$. As a consequence, there is no faithful $C^{1}$-action of $\mathit{BS}(1,n)$ on the torus with $h$ Anosov.

The rate of recurrence to measurable subsets in a conservative, ergodic infinite-measure-preserving system is quantified by generic divergence or convergence of certain sums given by a function $\omega (n)$. In the context of skew products over transformations of a probability space, we relate this notion to the more frequently studied question of the growth rate of ergodic sums (including Lyapunov exponents). We study in particular skew products over an irrational rotation given by bounded variation $ \mathbb{Z} $-valued functions: first the generic situation is studied and recurrence quantified, and then certain specific skew products over rotations are shown to violate this generic rate of recurrence.

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen–Margulis’ measure finiteness assumption used by Ricks [Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces. Ergod. Th. & Dynam. Sys. 37 (2017), 939–970] is removed. We also construct examples of CAT(0) spaces that do not admit finite Bowen–Margulis measure.

We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of $\mathbb{C}^2$. The first continuity result holds for any family of polynomial automorphisms of constant dynamical degree. The second result is the continuity of the upper exponent for families degenerating to a one-dimensional map.

Furstenberg’s $\times 2\times 3$ theorem asserts that the double sequence $(2^{m}3^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ is dense modulo one for every irrational $\unicode[STIX]{x1D6FC}$. The same holds with $2$ and $3$ replaced by any two multiplicatively independent integers. Here we obtain the same result for the sequences $((\begin{smallmatrix}m+n\\ d\end{smallmatrix})a^{m}b^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ for any non-negative integer $d$ and irrational $\unicode[STIX]{x1D6FC}$, and for the sequence $(P(m)a^{m}b^{n})_{m,n\geq 1}$, where $P$ is any polynomial with at least one irrational coefficient. Similarly to Furstenberg’s theorem, both results are obtained by considering appropriate dynamical systems.

Consider a class of Polish groups arising from the subclass of amenable locally compact ones via operations of countable projective limit and group extensions. We show that for each group from this class there exists a cocycle of an ergodic transformation with dense range in it. This is applied to extend and provide a short (orbital) proof for one of the main results from [ALV] on non-coalescence of some ergodic skew product extensions.