Let $f(z)=z^2+c$ be an infinitely renormalizable quadratic polynomial and $J_\infty $ be the intersection of forward orbits of ‘small’ Julia sets of its simple renormalizations. We prove that if f admits an infinite sequence of satellite renormalizations, then every invariant measure of $f: J_\infty \to J_\infty $ is supported on the postcritical set and has zero Lyapunov exponent. Coupled with [13], this implies that the Lyapunov exponent of such f at c is equal to zero, which partly answers a question posed by Weixiao Shen.

Let $(X,\mu ,T,d)$ be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for $\mu $-almost every $x\in X$,

$$ \begin{align*} \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \end{align*} $$

$\mu (B_k(x)) = M_k$

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.

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.

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.

In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim _{\mathbb {C}} M \geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal $\mathrm {U}(m)$-bundle $F_{\mathbb {C}}M$ of unitary frames. We show that if $m \geq 6$ is even and $m \neq 28$, there exists $\unicode{x3bb} (m) \in (0, 1)$ such that if $(M, g)$ has negative $\unicode{x3bb} (m)$-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $\unicode{x3bb} (m)$ satisfy $\unicode{x3bb} (6) = 0.9330...$, $\lim _{m \to +\infty } \unicode{x3bb} (m) = {11}/{12} = 0.9166...$, and $m \mapsto \unicode{x3bb} (m)$ is decreasing. This extends to the even-dimensional case the results of Brin and Gromov [On the ergodicity of frame flows. Invent. Math. 60(1) (1980), 1–7] who proved ergodicity of the unitary frame flow on negatively curved compact Kähler manifolds of odd complex dimension.

In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

We study infinite systems of mean field weakly coupled intermittent maps in the Pomeau–Manneville scenario. We prove that the coupled system admits a unique ‘physical’ stationary state, to which all absolutely continuous states converge. Moreover, we show that suitably regular states converge polynomially.

Let $S=\{p_1, \ldots , p_r,\infty \}$ for prime integers $p_1, \ldots , p_r.$ Let X be an S-adic compact nilmanifold, equipped with the unique translation-invariant probability measure $\mu .$ We characterize the countable groups $\Gamma $ of automorphisms of X for which the Koopman representation $\kappa $ on $L^2(X,\mu )$ has a spectral gap. More specifically, let Y be the maximal quotient solenoid of X (thus, Y is a finite-dimensional, connected, compact abelian group). We show that $\kappa $ does not have a spectral gap if and only if there exists a $\Gamma $-invariant proper subsolenoid of Y on which $\Gamma $ acts as a virtually abelian group,

We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure-preserving $\mathbb {Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a nullsequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third, and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on ${\mathbb Z}^{d}$-systems.

Let $\theta $ be an irrational real number. The map $T_\theta : y \mapsto (y+\theta ) \,\mod \!\!\: 1$ from the unit interval $\mathbf {I} = [0,1[$ (endowed with the Lebesgue measure) to itself is ergodic. In 2002, Rudolph and Hoffman showed in [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2) 156(1) (2002), 79–101] that the measure-preserving map $[T_\theta ,\mathrm {Id}]$ is isomorphic to a one-sided dyadic Bernoulli shift. Their proof is not constructive. A few years before, Parry [Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] had provided an explicit isomorphism under the assumption that $\theta $ is extremely well approached by the rational numbers, namely,

$$ \begin{align*}\inf_{q \ge 1} q^44^{q^2}~\mathrm{dist}(\theta,q^{-1}\mathbb{Z}) = 0.\end{align*} $$

Whether the explicit map considered by Parry is an isomorphism or not in the general case was still an open question. In Leuridan [Bernoulliness of $[T,\mathrm {Id}]$ when T is an irrational rotation: towards an explicit isomorphism. Ergod. Th. & Dynam. Sys. 41(7) (2021), 2110–2135] we relaxed Parry’s condition into

$$ \begin{align*}\inf_{q \ge 1} q^4~\mathrm{dist}(\theta,q^{-1}\mathbb{Z}) = 0.\end{align*} $$

In the present paper, we remove the condition by showing that the explicit map considered by Parry is always an isomorphism. With a few adaptations, the same method works with $[T,T^{-1}]$.

We study the measurable dynamical properties of the interval map generated by the model-case erasing substitution $\rho $, defined by

$$ \begin{align*} \rho(00)=\text{empty word},\quad \rho(01)=1,\quad \rho(10)=0,\quad \rho(11)=01. \end{align*} $$

A probability measure-preserving action of a discrete amenable group G is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for $G = \mathbb {Z}$, an action is dominant if and only if it has positive entropy and that for any G, positive entropy implies dominance. In this paper, we show that the converse also holds for any G, that is, that zero entropy implies non-dominance.

We provide an explicit $\mathcal {S}$-adic representation of rank-one subshifts with bounded spacers and call the subshifts obtained in this way ‘minimal Ferenczi subshifts’. We aim to show that this approach is very convenient to study the dynamical behavior of rank-one systems. For instance, we compute their topological rank, the strong and the weak orbit equivalence class. We observe that they have an induced system that is a Toeplitz subshift having discrete spectrum. We also characterize continuous and non-continuous eigenvalues of minimal Ferenczi subshifts.

We examine multiple ergodic averages of commuting transformations with polynomial iterates in which the polynomials may be pairwise dependent. In particular, we show that such averages are controlled by the Gowers–Host–Kra seminorms whenever the system satisfies some mild ergodicity assumptions. Combining this result with the general criteria for joint ergodicity established in our earlier work, we determine a necessary and sufficient condition under which such averages are jointly ergodic, in the sense that they converge in the mean to the product of integrals, or weakly jointly ergodic, in that they converge to the product of conditional expectations. As a corollary, we deduce a special case of a conjecture by Donoso, Koutsogiannis, and Sun in a stronger form.

An ergodic dynamical system $\mathbf {X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown by Glasner et al [On some generic classes of ergodic measure preserving transformations. Trans. Moscow Math. Soc. 82(1) (2021), 15–36] that Bernoulli systems with finite entropy are dominant. In this work, we show first that every ergodic system with positive entropy is dominant, and then that if $\mathbf {X}$ has zero entropy, then it is not dominant.

We generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg–Zimmer tower of a measurably distal system and showing that at each step there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

In this short and elementary note, we study some ergodic optimization problems for circle expanding maps. We first make an observation that if a function is not far from being convex, then its calibrated sub-actions are closer to convex functions in a certain effective way. As an application of this simple observation, for a circle doubling map, we generalize a result of Bousch saying that translations of the cosine function are uniquely optimized by Sturmian measures. Our argument follows the mainline of Bousch’s original proof, while some technical part is simplified by the observation mentioned above, and no numerical calculation is needed.