Let $ K $ be a compact subset of the d-torus invariant under an expanding diagonal endomorphism with s distinct eigenvalues. Suppose the symbolic coding of K satisfies weak specification. When $ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of $ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of $ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on $ K $ attains the Hausdorff dimension of $ K $. When $ s \geq 3 $, we find some examples in which statement (A) does not hold but statement (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of statements (A) and (B) for Bedford–McMullen sponges.

We define a family of discontinuous maps on the circle, called Bowen–Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor–Thurston theory for one-dimensional maps.

We prove the central limit theorem (CLT), the first-order Edgeworth expansion and a mixing local central limit theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise $C^2$ expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on $\mathbb {R}$. The class of observables in the CLT and the MLCLT on $\mathbb {R}$ include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber [Sampling the Lindelöf hypothesis with the Cauchy random walk. Proc. Lond. Math. Soc. (3) 98 (2009), 241–270] and Steuding [Sampling the Lindelöf hypothesis with an ergodic transformation. RIMS Kôkyûroku Bessatsu B34 (2012), 361–381] who have proven the strong law of large numbers for sampling the Lindelöf hypothesis.

Given two rational maps $f,g: \mathbb {P}^1 \to \mathbb {P}^1$ of degree d over $\mathbb {C}$, DeMarco, Krieger, and Ye [Common preperiodic points for quadratic polynomials. J. Mod. Dyn. 18 (2022), 363–413] have conjectured that there should be a uniform bound $B = B(d)> 0$ such that either they have at most B common preperiodic points or they have the same set of preperiodic points. We study their conjecture from a statistical perspective and prove that the average number of shared preperiodic points is zero for monic polynomials of degree $d \geq 6$ with rational coefficients. We also investigate the quantity $\liminf _{x \in \overline {\mathbb {Q}}} (\widehat {h}_f(x) + \widehat {h}_g(x) )$ for a generic pair of polynomials and prove both lower and upper bounds for it.

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

In this paper, we consider a class of affine Anosov mappings with a quasi-periodic forcing and show that there is a unique positive integer m, which only depends on the system, such that the exponential growth rate of the number of invariant tori of degree m is equal to the topological entropy.

We prove that among the set of pairs ($C^2$-diffeomorphism, $C^1$-potential), there exists a $C^1$-open and dense subset such that either the Lagrange spectrum is finite and the dynamics is a Morse–Smale diffeomorphism or the Lagrange spectrum has positive Hausdorff dimension and the system has positive topological entropy. We also prove that such dichotomy does not hold for typical systems when replacing the Lagrange by the Markov spectrum.

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As two applications, we first obtain the upcrossing inequalities with exponential decay of ergodic averages and then provide an explicit bound on the convergence rate such that the ergodic averages with strongly continuous regular group actions are metastable (or locally stable) on a large interval. Before exploiting the transference techniques, we actually obtain a stronger result—the jump estimates on a metric space with a measure not necessarily doubling. The ideas or techniques involve martingale theory, non-doubling Calderón–Zygmund theory, almost orthogonality argument, and some delicate geometric argument involving the balls and the cubes on a group equipped with a not necessarily doubling measure.

For full shifts on finite alphabets, Coelho and Quas [Criteria for $\overline {d}$-continuity. Trans. Amer. Math. Soc. 350(8) (1998), 3257–3268] showed that the map that sends a Hölder continuous potential $\phi $ to its equilibrium state $\mu _\phi $ is $\overline {d}$-continuous. We extend this result to the setting of full shifts on countable (infinite) alphabets. As part of the proof, we show that the map that sends a strongly positive recurrent potential to its normalization is continuous for potentials on mixing countable state Markov shifts.

In this work, we introduce the type and typeset invariants for equicontinuous group actions on Cantor sets; that is, for generalized odometers. These invariants are collections of equivalence classes of asymptotic Steinitz numbers associated to the action. We show the type is an invariant of the return equivalence class of the action. We introduce the notion of commensurable typesets and show that two actions which are return equivalent have commensurable typesets. Examples are given to illustrate the properties of the type and typeset invariants. The type and typeset invariants are used to define homeomorphism invariants for solenoidal manifolds.

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.

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 prove that in the space of $C^r$ maps $(r=2,\ldots ,\infty ,\omega )$ of a smooth manifold of dimension at least 4, there exist open regions where maps with infinitely many corank-2 homoclinic tangencies of all orders are dense. The result is applied to show the existence of maps with universal two-dimensional dynamics, that is, maps whose iterations approximate the dynamics of every map of a two-dimensional disk with an arbitrarily good accuracy. We show that maps with universal two-dimensional dynamics are $C^r$-generic in the regions under consideration.

We find necessary and sufficient conditions for high-order persistence of resonant caustics in perturbed circular billiards. The main tool is a perturbation theory based on the Bialy–Mironov generating function for convex billiards. All resonant caustics with period q persist up to order $\lceil q/n \rceil -1$ under any polynomial deformation of the circle of degree n.

We establish a polynomial ergodic theorem for actions of the affine group of a countable field K. As an application, we deduce—via a variant of Furstenberg’s correspondence principle—that for fields of characteristic zero, any ‘large’ set $E\subset K$ contains ‘many’ patterns of the form $\{p(u)+v,uv\}$, for every non-constant polynomial $p(x)\in K[x]$. Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a finitistic variant of Bergelson’s ‘colouring trick’, show that for $r\in \mathbb N$ fixed, any r-colouring of a large enough finite field will contain monochromatic patterns of the form $\{u,p(u)+v,uv\}$. In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalization of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic ‘colouring trick’, we provide a conditional, elementary generalization of Green and Sanders’ $\{u,v,u+v,uv\}$ theorem.

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space contains two disjoint subsets: one is a dense $G_\delta $ set for which all maximizing measures have ‘relatively small’ entropy; the other is the set of functions having uncountably many, fully supported ergodic maximizing measures with ‘relatively large’ entropy. This result generalizes and unifies the results of Morris [Discrete Contin. Dyn. Syst. 27 (2010), 383–388] and Shinoda [Nonlinearity 31 (2018), 2192–2200] on symbolic dynamics, and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without the Bowen specification property, including any transitive piecewise monotonic interval map, some coded shifts, and multidimensional $\beta $-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification.

We consider the Perron–Frobenius operator defined on the space of functions of bounded variation for the beta-map $\tau _\beta (x)=\beta x$ (mod $1$) for $\beta \in (1,\infty )$, and investigate its isolated eigenvalues except $1$, called non-leading eigenvalues in this paper. We show that the set of $\beta $ such that the corresponding Perron–Frobenius operator has at least one non-leading eigenvalue is open and dense in $(1,\infty )$. Furthermore, we establish the Hölder continuity of each non-leading eigenvalue as a function of $\beta $ and show in particular that it is continuous but non-differentiable, whose analogue was conjectured by Flatto, Lagarias and Poonen in [The zeta function of the beta transformation. Ergod. Th. & Dynam. Sys. 14 (1994), 237–266]. In addition, for an eigenfunctional of the Perron–Frobenius operator corresponding to an isolated eigenvalue, we give an explicit formula for the value of the functional applied to the indicator function of every interval. As its application, we provide three results related to non-leading eigenvalues, one of which states that an eigenfunctional corresponding to a non-leading eigenvalue cannot be expressed by any complex measure on the interval, which is in contrast to the case of the leading eigenvalue $1$.

In this work, we study rates of mixing for small independent and identically distributed random perturbations of contracting Lorenz maps sufficiently close to a Rovella parameter. By using a random Young tower construction, we prove that this random system has exponential decay of correlations.

Given a topologically transitive system on the unit interval, one can investigate the cover time, that is, the time for an orbit to reach a certain level of resolution in the repeller. We introduce a new notion of dimension, namely the stretched Minkowski dimension, and show that under mixing conditions, the asymptotics of typical cover times are determined by Minkowski dimensions when they are finite, or by stretched Minkowski dimensions otherwise. For application, we show that for countably full-branched affine maps, results using the usual Minkowski dimensions fail to give a finite limit of cover times, whilst the stretched version gives a finite limit. In addition, cover times for irrational rotations are calculated as counterexamples due to the absence of mixing.