We consider a simple model of an open partially expanding map. Its trapped set ${\mathcal{K}}$ in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component $\unicode[STIX]{x1D708}$ in the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call ‘minimal captivity’. This hypothesis is stable under perturbations and means that the dynamics is univalued in a neighborhood of ${\mathcal{K}}$. Under this hypothesis we show the existence of an asymptotic spectral gap and a fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit $\unicode[STIX]{x1D708}\rightarrow \infty$. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.

We construct the first examples of rational functions defined over a non-archimedean field with a certain dynamical property: the Julia set in the Berkovich projective line is connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we give an example for which the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.

In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $\unicode[STIX]{x1D719}$ is a Hölder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb{R}$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.

Let $G$ be a semisimple Lie group of rank one and $\unicode[STIX]{x1D6E4}$ be a torsion-free discrete subgroup of $G$. We show that in $G/\unicode[STIX]{x1D6E4}$, given $\unicode[STIX]{x1D716}>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than $\unicode[STIX]{x1D6FF}$ for a $1-\unicode[STIX]{x1D716}$ proportion of the time, for some $\unicode[STIX]{x1D6FF}>0$. The result also holds for any finitely generated discrete subgroup $\unicode[STIX]{x1D6E4}$ and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces. Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed $\unicode[STIX]{x1D716}>0$, there exists an injectivity radius $\unicode[STIX]{x1D6FF}$ such that, for any unipotent trajectory $\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$, either it spends at least a $1-\unicode[STIX]{x1D716}$ proportion of the time in the set with injectivity radius larger than $\unicode[STIX]{x1D6FF}$, for all large $T>0$, or there exists a $\{u_{t}\}_{t\in \mathbb{R}}$-normalized abelian subgroup $L$ of $G$ which intersects $g\unicode[STIX]{x1D6E4}g^{-1}$ in a small covolume lattice. We also extend these results to when $G$ is the product of rank-one semisimple groups and $\unicode[STIX]{x1D6E4}$ a discrete subgroup of $G$ whose projection onto each non-trivial factor is torsion free.

The evolution of cellular automata and of finitely anticipative transformations is studied by using right sets. These are the sets of symbols that are compatible with a past of a position and the respective coordinate of the transformation. Our main result shows, under some suitable conditions, that if the entropy converges to zero then the right sets increase towards the whole alphabet. We discuss these concepts with Wolfram automata.

Consider the skew product $F:\mathbb{T}^{2}\rightarrow \mathbb{T}^{2}$, $F(x,y)=(f(x),y+\unicode[STIX]{x1D70F}(x))$, where $f:\mathbb{T}^{1}\rightarrow \mathbb{T}^{1}$ is a piecewise $\mathscr{C}^{1+\unicode[STIX]{x1D6FC}}$ expanding map on a countable partition and $\unicode[STIX]{x1D70F}:\mathbb{T}^{1}\rightarrow \mathbb{R}$ is piecewise $\mathscr{C}^{1}$. It is shown that if $\unicode[STIX]{x1D70F}$ is not Lipschitz-cohomologous to a piecewise constant function on the joint partition of $f$ and $\unicode[STIX]{x1D70F}$, then $F$ is mixing at a stretched-exponential rate.

We consider the dynamics of holomorphic polynomials in $\mathbb{C}$. We show that the ergodic properties of the map can be seen already from the real parts of the orbits.

Let $\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>1$ and $T_{i}(x,y)=((x+i)/\unicode[STIX]{x1D6FD}_{1},(y+i)/\unicode[STIX]{x1D6FD}_{2}),i\in \{\pm 1\}$. Let $A:=A_{\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}}$ be the unique compact set satisfying $A=T_{1}(A)\cup T_{-1}(A)$. In this paper, we give a detailed analysis of $A$ and the parameters $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$ where $A$ satisfies various topological properties. In particular, we show that if $\unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FD}_{2}<1.202$, then $A$ has a non-empty interior, thus significantly improving the bound from Dajani et al [Self-affine sets with positive Lebesgue measure. Indag. Math. (N.S.)25 (2014), 774–784]. In the opposite direction, we prove that the connectedness locus for this family studied in Solomyak [Connectedness locus for pairs of affine maps and zeros of power series. Preprint, 2014, arXiv:1407.2563] is not simply connected. We prove that the set of points of $A$ which have a unique address has positive Hausdorff dimension for all $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$. Finally, we investigate simultaneous $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$-expansions of reals, which were the initial motivation for studying this family in Güntürk [Simultaneous and hybrid beta-encodings. Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference 2008, pp. 743–748].

We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex $S^{m-1}$. We prove that in contrast to the deterministic set-up the trajectories of the random dynamical system almost surely converge to one of the vertices of the simplex $S^{m-1}$, implying the survival of only one species. We also show that the minimal random point attractor of the system equals the set of all vertices. The convergence proof relies on a martingale-type limit theorem, which we prove in the appendix.

We determine limit distributions for return- and hitting-time functions of certain asymptotically rare events for conservative ergodic infinite measure preserving transformations with regularly varying asymptotic type. Our abstract result applies, in particular, to shrinking cylinders around typical points of null-recurrent renewal shifts and infinite measure preserving interval maps with neutral fixed points.

For every non-elementary hyperbolic group, we show that for every random walk with finitely supported admissible step distribution, the associated entropy equals the drift times the logarithmic volume growth if and only if the corresponding harmonic measure is comparable with Hausdorff measure on the boundary. Moreover, we introduce one parameter family of probability measures which interpolates a Patterson–Sullivan measure and the harmonic measure, and establish a formula of Hausdorff spectrum (multifractal spectrum) of the harmonic measure. We also give some finitary versions of dimensional properties of the harmonic measure.

We present new criteria, based on commutator methods, for the strong mixing property of discrete flows $\{U^{N}\}_{N\in \mathbb{Z}}$ and continuous flows $\{e^{-itH}\}_{t\in \mathbb{R}}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space ${\mathcal{H}}$. Our approach put into light a general definition for the topological degree of the maps $N\mapsto U^{N}$ and $t\mapsto e^{-itH}$ with values in the unitary group of ${\mathcal{H}}$. Among other examples, our results apply to skew products of compact Lie groups, time changes of horocycle flows and adjacency operators on graphs.

Let $X$ be a locally compact, $\unicode[STIX]{x1D70E}$-compact, non-compact Hausdorff space and let $\unicode[STIX]{x1D6FD}X$ be the Stone-Čech compactification of $X$. Let $G$ be a countably infinite discrete group continuously acting on $X$, and suppose that, for every $g\in G$, $\text{Fix}(g)=\{x\in X:gx=x\}$ is compact. The action of $G$ on $X$ induces the action on $\unicode[STIX]{x1D6FD}X$, and so on $X^{\ast }=\unicode[STIX]{x1D6FD}X\setminus X$. Let ${\mathcal{D}}$ denote the finest decomposition of $X^{\ast }$ into closed invariant subsets such that the corresponding quotient space of $X^{\ast }$ is Hausdorff. Such a decomposition can be defined for any action of $G$ on a compact Hausdorff space. Applying it to every member of ${\mathcal{D}}$ gives us a decomposition ${\mathcal{D}}^{2}$ of $X^{\ast }$, then ${\mathcal{D}}^{3}$, and so on. We show that (1) ${\mathcal{D}}^{\unicode[STIX]{x1D714}_{1}}$ is the coarsest decomposition of $X^{\ast }$ into closed invariant topologically transitive subsets, (2) there is a dense subset of points $p\in X^{\ast }$ such that $\overline{Gp}\in {\mathcal{D}}^{2}\setminus {\mathcal{D}}$, in particular, ${\mathcal{D}}$ is non-trivial and ${\mathcal{D}}^{2}$ is finer than ${\mathcal{D}}$, and (3) for every ordinal $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D714}_{1}$, there is $p\in X^{\ast }$ such that $\overline{Gp}\in {\mathcal{D}}^{\unicode[STIX]{x1D6FC}+2}\setminus {\mathcal{D}}^{\unicode[STIX]{x1D6FC}+1}$, so all the decompositions ${\mathcal{D}}^{\unicode[STIX]{x1D6FC}}$, $\unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D714}_{1}$, are distinct.