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 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.

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.

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 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 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 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.

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.

Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and have finite co-volume. A theorem of Lagarias [Meyer’s concept of quasicrystal and quasiregular sets. Comm. Math. Phys. 179(2) (1996), 365–376] provides a criterion for discrete subsets of Euclidean spaces to be approximate lattices. It asserts that if a subset X of $\mathbb {R}^n$ is relatively dense and $X - X$ is uniformly discrete, then X is an approximate lattice. We prove two generalizations of Lagarias’ theorem: when the ambient group is amenable and when it is a higher-rank simple algebraic group over a characteristic $0$ local field. This is a natural counterpart to the recent structure results for approximate lattices in non-commutative locally compact groups. We also provide a reformulation in dynamical terms pertaining to return times of cross-sections. Our method relies on counting arguments involving the so-called periodization maps, ergodic theorems and a method of Tao regarding small doubling for finite subsets. In the case of simple algebraic groups over local fields, we moreover make use of deep superrigidity results due to Margulis and to Zimmer.

We generalize Hopf’s theorem to thermostats: the total thermostat curvature of a thermostat without conjugate points is non-positive and vanishes only if the thermostat curvature is identically zero. We further show that, if the thermostat curvature is zero, then the flow has no conjugate points and the Green bundles collapse almost everywhere. Given a thermostat without conjugate points, we prove that the Green bundles are transverse everywhere if and only if it is projectively Anosov. Finally, we provide an example showing that Hopf’s rigidity theorem on the $2$-torus cannot be extended to thermostats. It is also the first example of a projectively Anosov thermostat which is not Anosov.

For upper semi-continuous potentials defined on shifts over countable alphabets, this paper ensures sufficient conditions for the existence of a maximizing measure. We resort to the concept of blur shift, introduced by T. Almeida and M. Sobottka as a compactification method for countable alphabet shifts consisting of adding new symbols given by blurred subsets of the alphabet. Our approach extends beyond the Markovian case to encompass more general countable alphabet shifts. In particular, we guarantee a convex characterization and compactness for the set of blur invariant probabilities with respect to the discontinuous shift map.

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.

Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the class whose number of asymptotic components is exactly the given cardinal. For finite or countable ones, we explicitly construct such examples using $\mathcal {S}$-adic subshifts. We obtain the uncountable case by showing that any topological dynamical system with at most countably many asymptotic components has zero topological entropy. We also construct systems that have arbitrarily high subexponential word complexity, but only one asymptotic component. We deduce that within any strong orbit equivalence class, there exists a minimal subshift whose automorphism group is isomorphic to $\mathbb {Z}$.

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu $ defined on the completion of the Borel $\sigma $-algebra, and $\mathcal F$ a f-invariant one-dimensional continuous foliation of M by $C^1$-leaves. Then, if f preserves a continuous $\mathcal {F}$-arc length system, then we only have three possibilities for the conditional measures of $\mu $ along $\mathcal F$, namely: (i) they are atomic for almost every leaf; or (ii) for almost every leaf, they are equivalent to the measure $\unicode{x3bb} _x$ induced by the invariant arc-length system over $\mathcal F$; or (iii) for almost every leaf, their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

We consider a DA-type surgery of the famous Lorenz attractor in dimension 4. This kind of surgery was first used by Smale [Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73(6) (1967), 747–817] and Mañé [Contributions to the stability conjecture. Topology 17(4) (1978), 383–396] to give important examples in the study of partially hyperbolic systems. Our construction gives the first example of a singular chain recurrence class which is Lyapunov stable, away from homoclinic tangencies, and exhibits robustly heterodimensional cycles. Moreover, the chain recurrence class has the following interesting property: there exists robustly a two-dimensional sectionally expanding subbundle (containing the flow direction) of the tangent bundle such that it is properly included in a subbundle of the finest dominated splitting for the tangent flow.