Let ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ where T is a maximal $K_{\mathcal {S}}$ -split torus in G. If $\# S = 2$ then the closure $ \overline{T\pi (g)}$ is a finite union of T-orbits stratified in terms of parabolic subgroups of ${\mathbf {G}} \times {\mathbf {G}}$ and, consequently, $\overline{T\pi (g)}$ is homogeneous (i.e. $\overline{T\pi (g)}= H\pi (g)$ for a subgroup H of G) if and only if ${T\pi (g)}$ is closed. On the other hand, if $\# \mathcal {S}> 2$ and K is not a $\mathrm {CM}$ -field then $\overline {T\pi (g)}$ is homogeneous for ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving (p.m.p.) actions of countablegroups introduced in Part I [B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310]. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semicontinuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.

We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $\operatorname {SL}(2, \mathbb {R})$ -action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $\operatorname {SL}(2, \mathbb {R})$ -action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the ‘weak convergence’ of push-forwards of horocycle measures under the geodesic flow and a short proof of weaker versions of theorems of Chaika and Eskin on Birkhoff genericity and Oseledets regularity in almost all directions for the Teichmüller geodesic flow.

We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.

We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one-dimensional torus translation, these estimates are nearly optimal.

A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

We show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.

This article investigates the long-time behavior of conservative affine processes on the cone of symmetric positive semidefinite $d\times d$ matrices. In particular, for conservative and subcritical affine processes we show that a finite $\log$ -moment of the state-independent jump measure is sufficient for the existence of a unique limit distribution. Moreover, we study the convergence rate of the underlying transition kernel to the limit distribution: first, in a specific metric induced by the Laplace transform, and second, in the Wasserstein distance under a first moment assumption imposed on the state-independent jump measure and an additional condition on the diffusion parameter.

A twisted cocycle taking values on a Lie group G is a cocycle that is twisted by an automorphism of G in each step. In the case where G = GL(d, ℝ), we prove that if two Hölder continuous twisted cocycles satisfying the so-called fiber-bunching condition have the same periodic data then they are cohomologous.

Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^{2}$-subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of $\mathbb{Z}^{2}$-subshifts with respect to a subaction of $\mathbb{Z}$. The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^{2}$-subshifts in terms of Kolmogorov–Sinai entropy.

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$-character variety of $\unicode[STIX]{x1D6F4}$. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$-maps $T_{\unicode[STIX]{x1D6FD}}$, for arbitrary $\unicode[STIX]{x1D6FD}>1$.

We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface. This answers a question posed in joint work by the first and the last named author with Shahar Mozes and Uri Shapira. Under certain congruence conditions we prove the joint equidistribution of conjugate rational points in the 2-torus and the modular surface.

In this article, we will prove a full topological version of Popa’s measurable cocycle superrigidity theorem for full shifts [Popa, Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups. Invent. Math. 170(2) (2007), 243–295]. Let $G$ be a finitely generated group that has one end, undistorted elements and sub-exponential divergence function. Let $H$ be a target group that is complete and admits a compatible bi-invariant metric. Then, every Hölder continuous cocycle for the full shifts of $G$ with value in $H$ is cohomologous to a group homomorphism via a Hölder continuous transfer map. Using the ideas of Behrstock, Druţu, Mosher, Mozes and Sapir [Divergence, thick groups, and short conjugators. Illinois J. Math. 58(4) (2014), 939–980; Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. Math. Ann. 344(3) (2009), 543–595; Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Amer. Math. Soc. 362(5) (2010), 2451–2505; Tree-graded spaces and asymptotic cones of groups. Topology 44(5) (2005), 959–1058], we show that the class of our acting groups is large including wide groups having undistorted elements and one-ended groups with strong thick of finite orders. As a consequence, irreducible uniform lattices of most of higher rank connected semisimple Lie groups, mapping class groups of $g$-genus surfaces with $p$-punches, $g\geq 2,p\geq 0$; Richard Thompson groups $F,T,V$; $\text{Aut}(F_{n})$, $\text{Out}(F_{n})$, $n\geq 3$; certain (two-dimensional) Coxeter groups; and one-ended right-angled Artin groups are in our class. This partially extends the main result in Chung and Jiang [Continuous cocycle superrigidity for shifts and groups with one end. Math. Ann. 368(3–4) (2017), 1109–1132].

We consider the dynamics of complex rational maps on $\widehat{\mathbb{C}}$. We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini–Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

We introduce inner amenability for discrete probability-measure-preserving (p.m.p.) groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations that either are stable or have a nontrivial central sequence in their full group.

We study the independence density for finite families of finite tuples of sets for continuous actions of discrete groups on compact metrizable spaces. We use it to show that actions with positive naive entropy are Li–Yorke chaotic and untame. In particular, distal actions have zero naive entropy. This answers a question of Lewis Bowen.

We consider dynamical systems $T:X\rightarrow X$ that are extensions of a factor $S:Y\rightarrow Y$ through a projection $\unicode[STIX]{x1D70B}:X\rightarrow Y$ with shrinking fibers, that is, such that $T$ is uniformly continuous along fibers $\unicode[STIX]{x1D70B}^{-1}(y)$ and the diameter of iterate images of fibers $T^{n}(\unicode[STIX]{x1D70B}^{-1}(y))$ uniformly go to zero as $n\rightarrow \infty$. We prove that every $S$-invariant measure $\check{\unicode[STIX]{x1D707}}$ has a unique $T$-invariant lift $\unicode[STIX]{x1D707}$, and prove that many properties of $\check{\unicode[STIX]{x1D707}}$ lift to $\unicode[STIX]{x1D707}$: ergodicity, weak and strong mixing, decay of correlations and statistical properties (possibly with weakening in the rates). The basic tool is a variation of the Wasserstein distance, obtained by constraining the optimal transportation paradigm to displacements along the fibers. We extend classical arguments to a general setting, enabling us to translate potentials and observables back and forth between $X$ and $Y$.

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if

$$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$

$[T_{\unicode[STIX]{x1D703}},\text{Id}]$

$\unicode[STIX]{x1D703}$

$[T_{\unicode[STIX]{x1D703}},\text{Id}]$

$$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$

$$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}|<+\infty ,\end{eqnarray}$$

$[0;a_{1},a_{2},\ldots ]$

$(p_{n}/q_{n})_{n\geq 0}$

$\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$

$\unicode[STIX]{x1D703}$

Von Neumann’s original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schrödinger and wave equations. In particular, decay estimates for time averages of solutions are shown.