In her thesis, Mirzakhani showed that the number of simple closed geodesics of length $\leq L$ on a closed, connected, oriented hyperbolic surface X of genus g is asymptotic to $L^{6g-6}$ times a constant depending on the geometry of X. In this survey, we give a detailed account of Mirzakhani’s proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces.

Using tools from computable analysis, we develop a notion of effectiveness for general dynamical systems as those group actions on arbitrary spaces that contain a computable representative in their topological conjugacy class. Most natural systems one can think of are effective in this sense, including some group rotations, affine actions on the torus and finitely presented algebraic actions. We show that for finitely generated and recursively presented groups, every effective dynamical system is the topological factor of a computable action on an effectively closed subset of the Cantor space. We then apply this result to extend the simulation results available in the literature beyond zero-dimensional spaces. In particular, we show that for a large class of groups, many of these natural actions are topological factors of subshifts of finite type.

For $E \subset \mathbb {N}$, a subset $R \subset \mathbb {N}$ is E-intersective if for every $A \subset E$ having positive relative density, $R \cap (A - A) \neq \varnothing $. We say that R is chromatically E-intersective if for every finite partition $E=\bigcup _{i=1}^k E_i$, there exists i such that $R\cap (E_i-E_i)\neq \varnothing $. When $E=\mathbb {N}$, we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when $E = \mathbb {P}$, the set of primes, or other sparse subsets of $\mathbb {N}$. Among other things, we prove the following: (1) the set of shifted Chen primes $\mathbb {P}_{\mathrm {Chen}} + 1$ is both intersective and $\mathbb {P}$-intersective; (2) there exists an intersective set that is not $\mathbb {P}$-intersective; (3) every $\mathbb {P}$-intersective set is intersective; (4) there exists a chromatically $\mathbb {P}$-intersective set which is not intersective (and therefore not $\mathbb {P}$-intersective).

In this paper, we study random walks on groups that contain superlinear-divergent geodesics, in the line of thoughts of Goldsborough and Sisto. The existence of a superlinear-divergent geodesic is a quasi-isometry invariant which allows us to execute Gouëzel’s pivoting technique. We develop the theory of superlinear divergence and establish a central limit theorem for random walks on these groups.

For a class of volume-preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of f is bounded above (respectively below) by the sum of the positive (respectively negative) Lyapunov exponents of its linearization. We show this for some classes of derived from Anosov (DA) and non-uniformly hyperbolic systems with dominated splitting, in particular for examples described by Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115(1) (2000), 157–193]. The results in this paper address a flexibility program by Bochi, Katok and Rodriguez Hertz [Flexibility of Lyapunov exponents. Ergod. Th. & Dynam. Sys. 42(2) (2022), 554–591].

Entropy of measure-preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are those given by the techniques developed by Ollagnier and Pinchon on the one hand and the Ornstein–Weiss lemma on the other. We extend these two approaches to the context of actions of amenable topological groups. In contrast to the discrete setting, our results reveal a remarkable difference between the two concepts of entropy in the realm of non-discrete groups: while the first quantity collapses to 0 in the non-discrete case, the second yields a well-behaved invariant for amenable unimodular groups. Concerning the latter, we moreover study the corresponding notion of topological pressure, prove a Goodwyn-type theorem, and establish the equivalence with the uniform lattice approach (for locally compact groups admitting a uniform lattice). Our study elaborates on a version of the Ornstein–Weiss lemma due to Gromov.

We study piecewise injective, but not necessarily globally injective, contracting maps on a compact subset of ${\mathbb R}^d$. We prove that, generically, the attractor and the set of discontinuities of such a map are disjoint, and hence the attractor consists of periodic orbits. In addition, we prove that piecewise injective contractions are generically topologically stable.

For given $\epsilon>0$ and $b\in \mathbb {R}^m$, we say that a real $m\times n$ matrix A is $\epsilon $-badly approximable for the target b if

$$ \begin{align*}\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b\rangle^m \geq \epsilon,\end{align*} $$

$\langle \cdot \rangle $

$\epsilon $

$\epsilon $

$\epsilon $

Let $f,g$ be $C^2$ expanding maps on the circle which are topologically conjugate. We assume that the derivatives of f and g at corresponding periodic points coincide for some large period N. We show that f and g are ‘approximately smoothly conjugate.’ Namely, we construct a $C^2$ conjugacy $h_N$ such that $h_N$ is exponentially close to h in the $C^0$ topology, and $f_N:=h_N^{-1}gh_N$ is exponentially close to f in the $C^1$ topology. Our main tool is a uniform effective version of Bowen’s equidistribution of weighted periodic orbits to the equilibrium state.

In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We look at the disintegration of transport plans, which are used to define and study disintegration maps. Using these objects, we study the regularity and absolute continuity of disintegration of measures. In particular, we exhibit conditions for which the disintegration map is weakly continuous and one can obtain a path of measures given by this map. We show a rigidity condition for the disintegration of measures to be given into absolutely continuous measures.