We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if $\Gamma $ is a countable discrete abelian group, $\varphi , \psi \in \mathrm {End}(\Gamma )$, and $\psi - \varphi $ is an injective endomorphism with finite index image, then for any ergodic measure-preserving $\Gamma $-system $( X, {\mathcal {X}}, \mu , (T_g)_{g \in \Gamma } )$, any measurable set $A \in {\mathcal {X}}$, and any ${\varepsilon }> 0$, there is a syndetic set of $g \in \Gamma$ such that $\mu ( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A ) > \mu(A)^3 - \varepsilon$. This generalizes the main results of Ackelsberg et al [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma 10 (2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma 10 (2022), Paper no. e107]. For the group $\Gamma = {\mathbb {Z}}^d$, the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma 10 (2022), Paper no. e107] that says that the relevant ergodic averages are controlled by a characteristic factor closely related to the quasi-affine (or Conze–Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to $\varphi $ and $\psi $) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum.

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.

We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.

We study actions of higher rank lattices $\Gamma <G$ on hyperbolic spaces and we show that all such actions satisfying mild properties come from the rank-one factors of G. In particular, all non-elementary isometric actions on an unbounded hyperbolic space are of this type.

In this paper, we study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology group has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than the modulus of the largest eigenvalue, then the map is robustly transitive. For this, we introduce Blichfedt’s theorem as a tool for extracting dynamical information from the action of a map in homology. We also treat the case of specially partially hyperbolic endomorphisms, for which we obtain a complete dichotomy: either the map is transitive and conjugated to its linear part, or its unstable foliation must contain an annulus which may either be wandering or periodic.

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.

Consider the quadratic family $T_a(x) = a x (1 - x)$ for $x \in [0, 1]$ and mixing Collet–Eckmann (CE) parameters $a \in (2,4)$. For bounded $\varphi $, set $\tilde \varphi _{a} := \varphi - \int \varphi \, d\mu _a$, with $\mu _a$ the unique acim of $T_a$, and put $(\sigma _a (\varphi ))^2 := \int \tilde \varphi _{a}^2 \, d\mu _a + 2 \sum _{i>0} \int \tilde \varphi _{a} (\tilde \varphi _{a} \circ T^i_{a}) \, d\mu _a$. For any mixing Misiurewicz parameter $a_{*}$, we find a positive measure set $\Omega _{*}$ of mixing CE parameters, containing $a_{*}$ as a Lebesgue density point, such that for any Hölder $\varphi $ with $\sigma _{a_{*}}(\varphi )\ne 0$, there exists $\epsilon _\varphi>0$ such that, for normalized Lebesgue measure on $\Omega _{*}\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$, the functions $\xi _i(a)=\tilde \varphi _a(T_a^{i+1}(1/2))/\sigma _a (\varphi )$ satisfy an almost sure invariance principle (ASIP) for any error exponent $\gamma>2/5$. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann’s proof for piecewise expanding maps. We need to introduce a variant of Benedicks–Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773–844].

Two asymptotic configurations on a full $\mathbb {Z}^d$-shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb {Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to $\mathbb {Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction.

Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9) 106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys. 43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with ‘pinched’ curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that, generically, the Birkhoff attractor is complicated, both from the topological and the dynamical points of view.

Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and $1$ Cantor winning in metric spaces, and the fact that $1/2$ winning implies absolute winning for subsets of $\mathbb {R}$. We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.

We prove that the existence of one horosphere in the universal cover of a closed Riemannian manifold of dimension $n \geq 3$ with strongly $1/4$-pinched or relatively $1/2$-pinched sectional curvature, on which the stable holonomy along one horosphere coincides with the Riemannian parallel transport, implies that the manifold is homothetic to a real hyperbolic manifold.

We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein’s $\bar {d}$ metric ($\bar {d}$-approachable shift spaces). The class of $\bar {d}$-approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $\bar {d}$-approachability, together with a closely connected notion of $\bar {d}$-shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys. 43(3) (2023), 943–970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, including the classic one and the almost/weak specification property, ensure $\bar {d}$-approachability and $\bar {d}$-shadowing. Here, we study further properties and connections between $\bar {d}$-shadowing and $\bar {d}$-approachability. We prove that $\bar {d}$-shadowing implies $\bar {d}$-stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the $\bar {d}$-shadowing property the Hausdorff pseudodistance ${\bar d}^{\mathrm {H}}$ between shift spaces induced by $\bar {d}$ is the same as the Hausdorff distance between their simplices of invariant measures with respect to the Hausdorff distance induced by Ornstein’s metric $\bar {d}$ between measures. We prove that without $\bar {d}$-shadowing this need not to be true (it is known that the former distance always bounds the latter). We provide examples illustrating these results, including minimal examples and proximal examples of shift spaces with the $\bar {d}$-shadowing property. The existence of such shift spaces was announced in the earlier paper mentioned above. It shows that $\bar {d}$-shadowing indeed generalizes the specification property.

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).

For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

We study the joint distribution of values of a pair consisting of a quadratic form ${\mathbf q}$ and a linear form ${\mathbf l}$ over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of $\mathrm{SL}_3(\mathbb{R})$. Math. Ann. 286 (1990), 101–128]. In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \ge 5$, then under the assumptions that for every $(\alpha , \beta ) \in {\mathbb {R}}^2 \setminus \{ (0,0) \}$, the form $\alpha {\mathbf q} + \beta {\mathbf l}^2$ is irrational and that the signature of the restriction of ${\mathbf q}$ to the kernel of ${\mathbf l}$ is $(p, n-1-p)$, where ${3\le p\le n-2}$, the number of vectors $v \in {\mathbb {Z}}^n$ for which $\|v\| < T$, $a < {\mathbf q}(v) < b$ and $c< {\mathbf l}(v) < d$ is asymptotically $ C({\mathbf q}, {\mathbf l})(d-c)(b-a)T^{n-3}$ as $T \to \infty $, where $C({\mathbf q}, {\mathbf l})$ only depends on ${\mathbf q}$ and ${\mathbf l}$. The density of the set of joint values of $({\mathbf q}, {\mathbf l})$ under the same assumptions is shown by Gorodnik [Oppenheim conjecture for pairs consisting of a linear form and a quadratic form. Trans. Amer. Math. Soc. 356(11) (2004), 4447–4463].

For an arbitrary countable discrete infinite group G, non-singular rank-one actions are introduced. It is shown that the class of non-singular rank-one actions coincides with the class of non-singular $(C,F)$-actions. Given a decreasing sequence of cofinite subgroups in G with $\bigcap _{n=1}^\infty \bigcap _{g\in G}g\Gamma _ng^{-1}=\{1_G\}$, the projective limit of the homogeneous G-spaces $G/\Gamma _n$ as $n\to \infty $ is a G-space. Endowing this G-space with an ergodic non-singular non-atomic measure, we obtain a dynamical system which is called a non-singular odometer. Necessary and sufficient conditions are found for a rank-one non-singular G-action to have a finite factor and a non-singular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one non-singular G-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the non-singular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the non-singular non-commutative case from the classical finite measure preserving one: odometer actions which are not of rank-one and factors of rank-one systems which are not of rank one; however, each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc.

Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and Hölder continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.

The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

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.