An account of the earliest known works on outer billiards is accompanied by a reproduction of the summary of B. H. Neumann’s 1959 colloquium and some highlights from the theory.

We consider non-autonomous conformal iterated function systems (NACIFSs) and their limit sets. Our main focus is on harmonic measure and its dimensions: Hausdorff and packing. We prove that these two dimensions are continuous under perturbations and that they satisfy Bowen’s and Manning’s type formulas. To achieve this, we establish general results about measures, and more broadly about positive functionals, defined on a symbolic space. We also develop tools from thermodynamic formalism in a non-autonomous setting.

We show that, under certain conditions, a strongly continuous semigroup admits an almost surely frequently hypercyclic random vector defined as a stochastic integral in Fréchet spaces with respect to the Brownian motion. Two criteria are given. We will apply the second criterion to three examples: translation semigroups on spaces of integrable functions, the exponential of weighted shifts, and the translation operators on the space of entire functions. This last example, with a stochastic approach, seems to be new in the literature. Some other examples are given.

To study any dynamical system, it is useful to find a partition that allows essentially faithful encoding (injective, up to a small exceptional set) into a subshift. Most topological and measure-theoretic systems can be represented by Bratteli–Vershik (or adic, or BV) systems. So, it is natural to ask when can a BV system be encoded essentially faithfully. Extending what was known previously for special diagrams and particular orders, we show here that for BV diagrams defined by homogeneous positive integer multivariable polynomials, and a wide family of their generalizations, which we call polynomial shape diagrams, for every choice of the edge ordering, the coding according to initial path segments of a fixed finite length is injective off of a negligible exceptional set.

The two-boost problem in space mission design asks whether two points of phase space can be connected with the help of two boosts of given energy. We provide a positive answer for a class of systems having a similar behaviour at infinity as the restricted three-body problem by defining and computing its Lagrangian Rabinowitz Floer homology. The principal technical challenge is dealing with the non-compactness of the associated energy hypersurfaces.

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.

Given $\beta>1$, let $T_\beta $ be the $\beta $-transformation on the unit circle $[0,1)$, defined by $T_\beta (x)=\beta x-\lfloor \beta x\rfloor $. For each $t\in [0,1)$, let $K_\beta (t)$ be the survivor set consisting of all $x\in [0,1)$ whose orbit $\{T^n_\beta (x): n\ge 0\}$ never enters the interval $[0,t)$. Kalle et al [Ergod. Th. & Dynam. Sys. 40(9) (2020), 2482–2514] considered the case $\beta \in (1,2]$. They studied the set-valued bifurcation set $\mathscr {E}_\beta :=\{t\in [0,1): K_\beta (t')\ne K_\beta (t)~\text { for all } t'>t\}$ and proved that the Hausdorff dimension function $t\mapsto \dim _H K_\beta (t)$ is a non-increasing Devil’s staircase. In a previous paper [Ergod. Th. & Dynam. Sys. 43(6) (2023), 1785–1828], we determined, for all $\beta \in (1,2]$, the critical value $\tau (\beta ):=\min \{t>0: \eta _\beta (t)=0\}$. The purpose of the present article is to extend these results to all $\beta>1$. In addition to calculating $\tau (\beta )$, we show that: (i) the function $\tau : \beta \mapsto \tau (\beta )$ is left-continuous on $(1,\infty )$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau $ has no downward jumps; and (iii) there exists an open set $O\subset (1,\infty )$, whose complement $(1,\infty )\setminus O$ has zero Hausdorff dimension, such that $\tau $ is real-analytic, strictly convex, and strictly decreasing on each connected component of O. We also prove several topological properties of the bifurcation set $\mathscr {E}_\beta $. The key to extending the results from $\beta \in (1,2]$ to all $\beta>1$ is an appropriate generalization of the Farey words that are used to parameterize the connected components of the set O. Some of the original proofs from the above-mentioned papers are simplified.

This paper is concerned with the study of random (Bernoulli and Markovian) products of matrices on a compact space of symbols. We establish the analyticity of the maximal Lyapunov exponent as a function of the transition probabilities, thus extending the results and methods of Y. Peres from a finite to an infinite (but compact) space of symbols. Our approach combines the spectral properties of the associated Markov operator with the theory of holomorphic functions in Banach spaces.

The density of a rational language can be understood as the frequency of some pattern in the shift space, for example, a pattern like ‘words with an even number of a given letter’. We study the density of group languages, that is, rational languages recognized by morphisms onto finite groups, inside shift spaces. We show that the density with respect to any given ergodic measure on a shift space exists for every group language, because it can be computed by using any ergodic lift of the given measure to a skew product between the shift space and the recognizing group. We then further study densities in shifts of finite type (with a suitable notion of irreducibility) and then in minimal shifts. In the latter case, we obtain a closed formula for the density under the condition that the aforementioned skew product has minimal closed invariant subsets that are ergodic under the product of the original measure and the uniform probability measure on the group. The formula is derived in part from a characterization of minimal closed invariant subsets for skew products between shifts and finite groups relying on notions of cocycles and coboundaries. In the case where the whole skew product is ergodic under the product measure, then the density is just the cardinality of the subset of the group that defines the language divided by the cardinality of the group. Moreover, we provide sufficient conditions for the skew product to have minimal closed invariant subsets that are ergodic under the product measure. Finally, we investigate the link between minimal closed invariant subsets, return words, and bifix codes.

We consider one-dimensional maps with several neutral fixed points that do not admit any physical measures. We show that there is a simplex of measures so that every measure in this simplex has a basin that has full Hausdorff dimension.

In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self-conformal/self-similar measures for iterated function systems satisfying the strong separation condition. We use these disintegration statements to prove new results on the Diophantine properties of these measures.

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.

We introduce a novel method for proving the ergodicity of skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel–Cantelli-type arguments given by Fayad and Lemańczyk [On the ergodicity of cylindrical transformations given by the logarithm. Mosc. Math. J. 6 (2006), 657–672]. The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and anti-symmetric cocycles. Moreover, its most significant advantage is the ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.

We characterize dynamics of every distortion element in the group of diffeomorphisms of the 2-sphere that has at least two fixed points and another recurrent point. The key result is that if f is such a diffeomorphism, then the homeomorphism $\check {f}_{\mathrm { ann}}$, which is a lift of the homeomorphism of the closed annulus $\overline {\mathcal {A}}$ obtained from $\mathbb {S}^2$ by blowing up two fixed points of f to the universal covering space of $\overline {\mathcal {A}}$, has a unique rotation number. Moreover, we find the differential of such a distortion element in the group of diffeomorphisms of the 2-sphere at each fixed point up to conjugacy.

We prove that when the Aubry set for a Lipschitz continuous potential is a subshift of finite type, then the pressure function converges exponentially fast to its asymptote as the temperature goes to 0. The speed of convergence turns out to be the unique eigenvalue for the matrix whose entries are the costs between the different irreducible pieces of the Aubry set. For a special case of Walters potential, we show that perturbations of that potential that go faster to zero than the pressure do not change the selection, neither for the subaction nor for the limit measure, a zero temperature.

We solve the dry ten Martini problem for the unitary almost-Mathieu operator with Diophantine frequencies in the non-critical regime.

We study random dynamical systems composed of Liverani–Saussol–Vaienti maps with varying parameters, without any mixing assumptions on the base space of random dynamics. We establish a quenched central limit theorem and identify conditions under which the associated limit variance varies continuously and differentiably with respect to perturbations of the random dynamics. Our arguments rely on recent results on statistical stability and linear response for random intermittent maps established by Dragičević et al [J. Lond. Math. Soc. 111 (2025), e70150].

We prove that the infinitely generated Apollonian gasket has full Hausdorff dimension spectrum. Our proof, which is computer assisted, relies on an iterative technique introduced by the first three authors [Chousionis, Leykekhman, and Urbański. The dimension spectrum of conformal graph directed Markov systems. Selecta Math. (N.S.) 25 (2019), 74] and on a flexible method for rigorously estimating Hausdorff dimensions of limit sets of conformal iterated function systems, which we recently developed [Chousionis et al. Rigorous Hausdorff dimension estimates for conformal fractals. Preprint, 2024, arXiv:2408.06330]. Another key ingredient in our proof is obtaining, for the first time, reasonably sized distortion constants for the (infinite) Apollonian iterated function system.

Consider the four-punctured sphere $\mathbb {S}_4^2$. Each choice of four traces, one for each puncture, determines a relative character variety for the representations of the fundamental group of $\mathbb {S}_4^2$ in ${\textsf {SL}}_2(\mathbb {C})$. We classify the stationary probability measures for the action of the mapping class group ${\textsf {Mod}}(\mathbb {S}_4^2)$ on these character varieties.