Let $D^{-}$ and $D^{+}$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\rightarrow +\infty$ for the number of common perpendiculars of length at most $t$ from $D^{-}$ to $D^{+}$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^{-}$ and $D^{+}$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic formula for the number of connected components of the domain of discontinuity of Kleinian groups as their diameter goes to $0$.

Let $X_{0}$ be a smooth projective threefold which is Fano or which has Picard number 1. Let $\unicode[STIX]{x1D70B}:X\rightarrow X_{0}$ be a finite composition of blowups along smooth centers. We show that for ‘almost all’ of such $X$, if $f\in \text{Aut}(X)$, then its first and second dynamical degrees are the same. We also construct many examples of blowups $X\rightarrow X_{0}$, on which any automorphism is of zero entropy. The main idea is that, because of the log-concavity of dynamical degrees and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology classes. We will also discuss a possible application of these results to a threefold constructed by Kenji Ueno.

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and non-dense forward orbits under the other is a dense, uncountable set. The pair of maps can be non-commuting. We also show the same for pairs of $C^{2}$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.

We consider the Lagrange and the Markov dynamical spectra associated to horseshoes on a surface with Hausdorff dimension greater than one. We show that for a ‘large’ set of real functions on the surface and for ‘typical’ horseshoes with Hausdorff dimension greater than one, both the Lagrange and the Markov dynamical spectra have persistently non-empty interior.

In this paper we observe that one of our main results in ‘Optimal transport and dynamics of circle expanding maps acting on measures’ [Ergod. Th. & Dynam. Sys.33(2) (2013), 529–548] has an interesting consequence: an infinitesimal version of the Furstenberg conjecture is false in a very strong way. More precisely, we find deformations of the Lebesgue measure on the circle which are first-order invariant simultaneously for all integer multiplications modulo 1. We also correct an error in a lemma of the mentioned article. Both the proof and the statement must be corrected, but the main results of the article are not affected.

In this paper, we study the dynamics of the family of rational maps with two parameters

$$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$

$n\geq 2$

$a,b\in \mathbb{C}^{\ast }$

$f_{a,b}$

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.

Let $X$ be a proper, geodesically complete CAT($0$) space under a proper, non-elementary, isometric action by a group $\unicode[STIX]{x1D6E4}$ with a rank one element. We construct a generalized Bowen–Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $\unicode[STIX]{x1D6E4}$-action. Although the construction of Bowen–Margulis measures for rank one non-positively curved manifolds and for CAT($-1$) spaces is well known, the construction for CAT($0$) spaces hinges on establishing a new structural result of independent interest: almost no geodesic, under the Bowen–Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in $\unicode[STIX]{x2202}_{\infty }X$, under the Patterson–Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen–Margulis measure is finite, as it is in the cocompact case.) Finally, we precisely characterize mixing when $X$ has full limit set: a finite Bowen–Margulis measure is not mixing under the geodesic flow precisely when $X$ is a tree with all edge lengths in $c\mathbb{Z}$ for some $c>0$. This characterization is new, even in the setting of CAT($-1$) spaces. More general (technical) versions of these results are also stated in the paper.

We use the homotopy Brouwer theory of Handel to define a Poincaré index between pairs of orbits for an orientation-preserving fixed-point-free homeomorphism of the plane. Furthermore, we prove that this index is almost additive.

Let $f:M\rightarrow M$ be a biholomorphism on a two-dimensional complex manifold, and let $X\subseteq M$ be a compact $f$-invariant set such that $f|_{X}$ is asymptotically dissipative and without periodic sinks. We introduce a solely dynamical obstruction to dominated splitting, namely critical point. Critical point is a dynamical object and captures many of the dynamical properties of a one-dimensional critical point.

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.

We characterize supramenable groups in terms of the existence of invariant probability measures for partial actions on compact Hausdorff spaces and the existence of tracial states on partial crossed products. These characterizations show that, in general, one cannot decompose a partial crossed product of a $\text{C}^{\ast }$-algebra by a semidirect product of groups into two iterated partial crossed products. However, we give conditions which ensure that such decomposition is possible.

Let $\unicode[STIX]{x1D714}=[a_{1},a_{2},\ldots ]$ be the infinite expansion of a continued fraction for an irrational number $\unicode[STIX]{x1D714}\in (0,1)$, and let $R_{n}(\unicode[STIX]{x1D714})$ (respectively, $R_{n,k}(\unicode[STIX]{x1D714})$, $R_{n,k+}(\unicode[STIX]{x1D714})$) be the number of distinct partial quotients, each of which appears at least once (respectively, exactly $k$ times, at least $k$ times) in the sequence $a_{1},\ldots ,a_{n}$. In this paper, it is proved that, for Lebesgue almost all $\unicode[STIX]{x1D714}\in (0,1)$ and all $k\geq 1$,

$$\begin{eqnarray}\displaystyle \lim _{n\rightarrow \infty }\frac{R_{n}(\unicode[STIX]{x1D714})}{\sqrt{n}}=\sqrt{\frac{\unicode[STIX]{x1D70B}}{\log 2}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n}(\unicode[STIX]{x1D714})}=\frac{C_{2k}^{k}}{(2k-1)\cdot 4^{k}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n,k+}(\unicode[STIX]{x1D714})}=\frac{1}{2k}.\end{eqnarray}$$

$R_{n}$

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.

The Kari–Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other examples of aperiodic Wang tiles. We show that the $\mathbb{Z}^{2}$ action by translation on a certain subset of the Kari–Culik tilings, namely those whose rows can be interpreted as Sturmian sequences (rotation sequences), is minimal. We give a characterization of this space as a skew product as well as explicit bounds on the waiting time between occurrences of $m\times n$ configurations.

We apply coupling techniques in order to prove that the transfer operators associated with random topological Markov chains and non-stationary shift spaces with the big images and preimages property have a spectral gap.

We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.