Let $\gamma _{n}= O (\log ^{-c}n)$ and let $\nu $ be the infinite product measure whose nth marginal is Bernoulli $(1/2+\gamma _{n})$. We show that $c=1/2$ is the threshold, above which $\nu $-almost every point is simply Poisson generic in the sense of Peres and Weiss, and below which this can fail. This provides a range in which $\nu $ is singular with respect to the uniform product measure, but $\nu $-almost every point is simply Poisson generic.

We show that for a minimal system $(X,T)$, the set of saturated points along cubes with respect to its maximal $\infty $-step pro-nilfactor $X_\infty $ has a full measure. As an application, it is shown that if a minimal system $(X,T)$ has no non-trivial $(k+1)$-tuples with arbitrarily long finite IP-independence sets, then it has only at most k ergodic measures and is an almost $k'$ to one extension of $X_\infty $ for some $k'\leqslant k$. In particular, for $k=1$, we prove that $(X,T)$ is uniquely ergodic (even regular with respect to $X_\infty $), which answers a conjecture stated by Dong et al [Infinite-step nilsystems, independence and complexity. Ergod. Th. & Dynam. Sys. 33(1) (2013), 118–143].

The marked length spectrum (MLS) of a closed negatively curved manifold $(M, g)$ is known to determine the metric g under various circumstances. We show that, in these cases, (approximate) values of the MLS on a sufficiently large finite set approximately determine the metric. Our approach is to recover the hypotheses of our main theorems in Butt [Quantative marked length spectrum rigidity. Preprint, 2022], namely, multiplicative closeness of the MLS functions on the entire set of closed geodesics of M. We use mainly dynamical tools and arguments, but take great care to show that the constants involved depend only on concrete geometric information about the given Riemannian metrics, such as the dimension, diameter and sectional curvature bounds.

We compute the Euler characteristic of the moduli space of quadratic rational maps with a periodic marked critical point of a given period.

We examine the convergence of ergodic averages along polynomials in Toeplitz systems and prove that it is possible for averages along one polynomial to converge, and along another to diverge. We also study the density of the polynomial orbits in Toeplitz systems—we show that it implies equidistribution of the polynomial orbits in the class of regular Toeplitz systems, but not in the class of strictly ergodic ones.

This is a corrigendum to ‘An uncountable Furstenberg–Zimmer structure theory’ [Ergod. Th. & Dynam. Sys. 43(7) (2023), 2404–2436]. We report two issues in that paper. First, Lemma A.5 and Proposition A.6 in the Appendix, which supported a spectral analysis of conditional Hilbert–Schmidt operators, are incorrect. These results were used in the proof of Lemma 4.4, which establishes part of the equivalences in Theorem 4.1. We provide a correction for this issue here. While the proof strategy of Lemma 4.4 remains valid, the details have been revised using known auxiliary results in the non-commutative setting of tracial von Neumann algebras, replacing the faulty arguments from the Appendix. Second, the proof of the implication $\mathrm{(iii)} \Rightarrow \mathrm{(iii)}'$ in Lemma 4.10 is incorrect. We supply a new argument to address this. We also take this opportunity to correct several minor issues that have come to our attention since the paper’s publication. A fully revised version, including these corrections, as well as updated references and some fixed typos, is now available on arXiv.

We establish large deviation estimates related to the Darling–Kac theorem and generalized arcsine laws for occupation and waiting times of ergodic transformations preserving an infinite measure, such as non-uniformly expanding interval maps with indifferent fixed points. For the proof, we imitate the study of generalized arcsine laws for occupation times of one-dimensional diffusion processes and adopt a method of double Laplace transform.

We show that the continuity property of Lyapunov exponents proved by Buzzi et al. [Continuity properties of Lyapunov exponents for surface diffeomorphisms. Invent. Math. 230(2) (2022), 767–849] for smooth surface diffeomorphisms extends to smooth interval maps in the case when the map only has non-flat critical points and the entropies converging to the topological entropy. The result we obtained is stronger than the continuity of Lyapunov exponents, in particular, we prove the uniform integrability of Lyapunov exponents over entropies.

Let $\Omega \in \mathbb C$ be a domain such that $K:= \mathbb C \setminus \Omega $ is compact and non-polar. Let $(q_k)_{k>0}$ be a sequence of polynomials with $n_k$, the degree of $q_k$ satisfying $n_k \to \infty $, and let $(q_k^{(m)})_k$ denote the sequence of mth derivatives. We provide conditions which ensure that the preimages $(q_k^{(m)})^{-1}(\{a\})$ uniformly equidistribute on $\partial \Omega $, as $k \to \infty $, for every $a \in \mathbb C$ and every $m = 0, 1, \ldots .$

In this paper, we prove a quantitative equidistribution theorem for polynomial sequences in a nilmanifold, where the average is taken along spheres instead of cubes. To be more precise, let $\Omega \subseteq \mathbb {Z}^{d}$ be the preimage of a sphere $\mathbb {F}_{p}^{d}$ under the natural embedding from $\mathbb {Z}^{d}$ to $\mathbb {F}_{p}^{d}$. We show that if a rational polynomial sequence $(g(n)\Gamma )_{n\in \Omega }$ is not equidistributed on a nilmanifold $G/\Gamma $, then there exists a non-trivial horizontal character $\eta $ of $G/\Gamma $ such that $\eta \circ g \,\mod \mathbb {Z}$ vanishes on $\Omega $.

We prove that every ergodic transformation is Shannon orbit equivalent to a weakly mixing transformation. The proof is based on the techniques introduced by Fieldsteel and Friedman to show that there is a mixing transformation for a given ergodic transformation T that is, for all $a\geq 1$, weak-a-equivalent to T and, for all ${b\in (0,1)}$, strong-b-equivalent to T. In particular, we adapt the construction of Fieldsteel and Friedman by which they permute the columns of each Rokhlin tower in a sequence of rapidly growing Rokhlin towers so that the corresponding cocycles converge to an orbit equivalence cocycle of T such that the resulting transformation and orbit equivalence have the desired properties. In addition to this, we demonstrate a flexible method for obtaining actions of $\mathbb {Z}^{2} $ that are Shannon orbit equivalent to a given ergodic transformation.

We study the the asymptotic dynamics of elementary cellular automaton 18 through its limit set, generic limit set and $\mu $-limit set. The dynamics of rule 18 are characterized by persistent local patterns known as kinks. We characterize the configurations of the generic limit set containing at most two kinks. As a corollary, we show that the three limit sets of rule 18 are distinct.

We investigate the thermodynamic formalism for Viana maps—skew products obtained by coupling an expanding circle map with a slightly perturbed quadratic family on the fibers. For every Hölder potential $\varphi $ whose oscillation is below an explicit threshold, we show that an equilibrium state not only exists, but is unique and satisfies an upper level-2 large-deviation principle. All of these conclusions persist under sufficiently small perturbations of the reference map.

In this paper, we prove that for every $C^1$ star vector field on three-dimensional manifolds, every ergodic hyperbolic invariant measure which is not supported on singularities can be approximated by periodic measures and that the Lyapunov exponents of the ergodic hyperbolic invariant measure can also be approximated by the Lyapunov exponents of those periodic measures.

Let f be a rational map of degree $d\geq 2$. The moduli space $\mathcal {M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting of all quasiconformal conjugacy classes of f. For f that is not flexible Lattès, we show that there is a normal affine variety $X_f$ of dimension $2d-2$ and a holomorphic injection $i:\mathcal {M}_f\to X_f$ such that $i(\mathcal {M}_f)$ is precompact in $X_f$. In particular, $\mathcal {M}_f$ is Carathéodory hyperbolic (that is, bounded holomorphic functions separate points in $\mathcal {M}_f$), provided that f is not flexible Lattès. This solves a conjecture of McMullen. When $d\geq 4$, we give a concrete construction of $X_f$ as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism.

In this paper, it is shown that for every lattice $\Gamma \subset PSL_2(\mathbb {R})$, there exists a $c>0$ such that for any $0 \leq \gamma <c$, the sequence $p h(n^{1+\gamma })$ equidistributes for any $p \in \Gamma \backslash PSL_2(\mathbb {R})$, where h is the horocycle flow. This makes modest progress towards a conjecture of Shah and generalizes a result of Venkatesh [Sparse equidistribution problems, period bounds, and subconvexity. Ann. of Math. (2) 172(2) (2010), 989–1094], who established the same equidistribution for co-compact lattices. The proof uses a dichotomy between good equidistribution estimates and approximability of $\{p h(t), t \leq T \}$ by closed horocycles of small period.

We study the ergodic properties of the translation surface $X_{\unicode{x3bb} ,\mu }$ formed by gluing two flat tori along a slit with holonomy $(\unicode{x3bb} ,\mu ) \in \mathbb {R}^2$. Extending the dichotomy result of Cheung, Hubert, and Masur for the case $\mu = 0$, we prove the following: for slits not parallel to any absolute homology class, the Hausdorff dimension of the set $\operatorname {\mathrm {NE}}(X_{\unicode{x3bb} ,\mu },\omega )$ of non-ergodic directions is either $0$ or $\frac 12$. This dichotomy is completely characterized by the Pérez–Marco condition expressed in terms of best approximation denominators. As a corollary, we obtain that the Pérez–Marco condition for best approximation denominators is norm-independent.

We study piecewise quasiconformal covering maps of the unit circle. We provide sufficient conditions so that a conjugacy between two such dynamical systems has a quasiconformal or David extension to the unit disk. Our main result generalizes the main result of Lyubich et al [David extension of circle homeomorphisms, welding, mating, and removability. Mem. Amer. Math. Soc. 313 (2025), 1588], which deals with piecewise analytic maps. As applications, we provide a classification of piecewise quasiconformal maps of the circle up to quasisymmetric conjugacy, we prove a general conformal mating theorem for Blaschke products and we study the quasiconformal geometry of parabolic basins.

For an arbitrary negative Schwarzian unimodal map with a non-flat critical point, we establish the level-2 large deviation principle for empirical distributions. We also give an example of a bimodal map for which the level-2 large deviation principle does not hold.

In this article, we prove that the set of well-approximable points $W_\varphi (z) = \{x \in X : d (f^n x, z ) < \varphi (n) \mathrm {\ for\ infinite\ } n \in \mathbb {N}^+\}$ in the shrinking targets problems is distributional chaotic of type 1 for systems with a weak form of the exponential specification property. We apply it to transitive Anosov systems, $\beta $-shifts, etc.