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 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 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.

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 construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop the theory and use it in particular to provide examples of thick metric attractors with intermingled basins.

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 $m = 0, 1, \ldots .$

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.

We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we obtain a general closed form expression for the $L^q$-spectra. Consequently, we obtain a closed form expression for box dimensions of associated planar graph-directed box-like self-affine sets. We also provide a precise answer to a question posed by Fraser [On the $L^q$-spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368(8) (2016), 5579–5620] concerning the $L^q$-spectra of planar self-affine measures generated by diagonal matrices. An interesting observation of the closed form expression is that it is possible to calculate the $L^q$-spectrum of a measure without involving the exact $L^q$-spectra of its projections to the axes.