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 a generic probability measure-preserving (p.m.p.) action of a countable amenable group G has scaling entropy that cannot be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of G for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. G-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that cannot be bounded from below by a given sequence. In addition, we show an example of an amenable group that has such a lower bound for every free p.m.p. action.

We prove that if two free probability-measure-preserving (p.m.p.) ${\mathbb Z}$-actions are Shannon orbit equivalent, then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein–Weiss and the entropy invariance results of Austin and Kerr–Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also show, at the opposite end of the stochastic spectrum, that every ${\mathbb Z}$-odometer is Shannon orbit equivalent to the universal ${\mathbb Z}$-odometer.

The first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theorem for the associated random processes, we discover some rigid relationships between derivatives of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of Hölder potentials globally, which answers a question of Kucherenko and Quas. In the second part of the work, we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as the number of symbols in the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.

We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.

Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties, such as an infimum rule. Next, we extend the definitions of different notions of Gibbs measures and prove their existence and equivalence, given some regularity and normalization criteria on the potential. Finally, we provide a family of potentials that nontrivially satisfy the conditions for having this equivalence and a nonempty range of inverse temperatures where uniqueness holds.

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

Based on previous work of the authors, to any S-adic development of a subshift X a ‘directive sequence’ of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S-adic development. The issuing rich picture enables one to deduce results about X with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result, we also exhibit, for any integer $d \geq 2$, an S-adic development of a minimal, aperiodic, uniquely ergodic subshift X, where all level alphabets $\mathcal A_n$ have cardinality $d,$ while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subseteq \mathcal A_n^{\mathbb {Z}}$.

We prove that, for any countable acylindrically hyperbolic group G, there exists a generating set S of G such that the corresponding Cayley graph $\Gamma (G,S)$ is hyperbolic, $|\partial \Gamma (G,S)|>2$, the natural action of G on $\Gamma (G,S)$ is acylindrical and the natural action of G on the Gromov boundary $\partial \Gamma (G,S)$ is hyperfinite. This result broadens the class of groups that admit a non-elementary acylindrical action on a hyperbolic space with a hyperfinite boundary action.

For $\mathscr {B} \subseteq \mathbb {N} $, the $ \mathscr {B} $-free subshift $ X_{\eta } $ is the orbit closure of the characteristic function of the set of $ \mathscr {B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\eta } $, have their analogues for $ X_{\eta } $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures [G. Keller. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21(3) (2021), Paper No. 2140008]. A central assumption in our work is that $\eta ^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_{\eta } $) is regular. From this, we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_{\eta } $ from above and below.

For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

We describe two kinds of regular invariant measures on the boundary path space $\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C$^{*}$-algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on $C^{*}(E)$ are gauge-invariant when $C^{*}(E)$ is simple and separable.

Using Patterson–Sullivan measures, we investigate growth problems for groups acting on a metric space with a strongly contracting element.

We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3) 101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.

We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.

Let $ G $ be a connected semisimple real algebraic group and $\Gamma <G$ be a Zariski dense discrete subgroup. Let N denote a maximal horospherical subgroup of G, and $P=MAN$ the minimal parabolic subgroup which is the normalizer of N. Let $\mathcal E$ denote the unique P-minimal subset of $\Gamma \backslash G$ and let $\mathcal E_0$ be a $P^\circ $-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary $ G/P $ and show that the following are equivalent for any $[g]\in \mathcal E_0$:

1. (1) $gP\in G/P$ is a horospherical limit point;

2. (2) $[g]NM$ is dense in $\mathcal E$;

3. (3) $[g]N$ is dense in $\mathcal E_0$.

The equivalence of items (1) and (2) is due to Dal’bo in the rank one case. We also show that unlike convex cocompact groups of rank one Lie groups, the $NM$-minimality of $\mathcal E$ does not hold in a general Anosov homogeneous space.

In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $. We consider a class of non-square-integrable observables $\phi $, mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$, where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$. The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,\ldots , f_k$ such that the functions $f_i\circ T^n$, $n\in {\mathbb {Z}}$, $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.

We show that stationary time series can be uniformly approximated over all finite time intervals by mixing, non-ergodic, non-mean-ergodic, and periodic processes, and by codings of aperiodic processes. A corollary is that the ergodic hypothesis—that time averages will converge to their statistical counterparts—and several adjacent hypotheses are not testable in the non-parametric case. Further Baire category implications are also explored.

We show a new method of estimating the Hausdorff measure of a set from below. The method requires computing the subsequent closest return times of a point to itself.