Let $m\in \mathbb{N}$ and $\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$ be a measure-preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $\unicode[STIX]{x1D708}$ on $\mathbb{R}^{m}$ is weakly equidistributed for $\mathbf{X}$ if there exists $A\subseteq \mathbb{R}$ of density 1 such that, for all $f\in L^{\infty }(\unicode[STIX]{x1D707})$, we have

$$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$W(\mathbf{X})$

$\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$

$\mathbb{R}$

$(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$

$\unicode[STIX]{x1D708}$

$\mathbb{R}^{m}$

$\mathbf{X}$

$\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$

$\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$

$\unicode[STIX]{x1D708}$

$\mathbb{R}^{m}$

$\unicode[STIX]{x1D708}(\ell )=0$

$\ell$

$\mathbb{R}^{m}$

A probability measure-preserving action of a discrete amenable group G is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for $G = \mathbb {Z}$, an action is dominant if and only if it has positive entropy and that for any G, positive entropy implies dominance. In this paper, we show that the converse also holds for any G, that is, that zero entropy implies non-dominance.

We show that for almost every $(P,\unicode[STIX]{x1D706})$, where $P$ is a convex polygon and $\unicode[STIX]{x1D706}\in (0,1)$, the corresponding outer billiard about $P$ with contraction $\unicode[STIX]{x1D706}$ is asymptotically periodic, i.e., has a finite number of periodic orbits and every orbit is attracted to one of them.

In this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

We show some level-2 large deviation principles for real and complex one-dimensional maps satisfying a weak form of hyperbolicity. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages.

We show that near the geodesic flow of a compact Riemannian manifold with no conjugate points which is expansive, every expansive geodesic flow has no conjugate points. We also prove that in the above hypotheses the geodesic flow istopologically stable.

We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. We also give conditions for expansivity of the quotient, and applications to combinatorial group theory, knot theory and topological quantum field theory.

We prove a meromorphic integrability criterion for the geodesic flow of an algebraic manifold of the form ${z}^{p} - f({x}_{1} , \ldots , {x}_{n} )= 0$ with the induced metric of ${ \mathbb{C} }^{n+ 1} $ and $f$ a homogeneous rational function, using a parallel between the properties of such algebraic manifolds and homogeneous potentials. We then apply this criterion to the manifolds of the form $z= {\lambda }_{1} { x}_{1}^{k} + \cdots + {\lambda }_{n} { x}_{n}^{k} $, $k\in { \mathbb{Z} }^{+ } $, and ${x}^{n} {y}^{m} {z}^{l} = 1, n, m, l\in \mathbb{Z} $, and prove that their geodesic flow is not integrable except for some given exceptional cases.

Let \{U_{g}\}_{g\in G} be a weakly mixing unitary action of a finitely-generated nilpotent group on a Hilbert space \mathcal{H}. Extending a known result about abelian groups, we show that weakly wandering vectors are dense in \mathcal{H}.

We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of ‘symmetric measure’: exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one-dimensional case) exhibit a variety of spectral properties.

We prove that a nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{R}^{2}$ preserving a compact set has a global fixed point. These results generalize theorems of Franks et al for the abelian case. We show that a nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$ that has a finite orbit of odd cardinality also has a global fixed point. Moreover, we study the properties of the 2-points orbits of nilpotent fixed-point-free subgroups of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$.

We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $\mathbb {R}$ -actions, coming from a single jointly measurable measure-preserving $\mathbb {R}^2$ -action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.

Let M={\tilde M}/\Gamma be a closed negatively curved manifold with universal covering {\tilde M} and fundamental group \Gamma. Every Gibbs equilibrium state \nu of a Hölder continuous function on the unit tangent bundle T^1M of M projects to a \Gamma-invariant ergodic measure class mc(\nu_+) on the ideal boundary \partial{\tilde M} of {\tilde M}. We show that this measure class is also ergodic under the action of any normal subgroup \Gamma^\prime of \Gamma for which the factor group \Gamma/\Gamma^\prime is nilpotent.

Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

We show that in the context of homogeneous Cantor sets, there are generically five possible (open and dense) structures for their arithmetic sum: a Cantor set, an L, R, M-Cantorval and a finite union of closed intervals. The dense case has been dealt with previously. In this paper, we explicitly present pairs of this space which have stable intersection, while not satisfying the generalized thickness test. Also, all the pairs of middle homogeneous Cantor sets whose arithmetic sum is a closed interval are identified.

We prove that the pressure metric on the Teichmüller space of a bordered surface is incomplete and that a completion can be given by the moduli space of metrics on a graph (dual to a special ideal triangulation of the same bordered surface) equipped with pressure metric. In contrast to the closed surface case, we obtain as a corollary that the pressure metric is not bi-Lipschitz to the Weil–Petersson metric.

We study cohomology-free (c.f.) diffeomorphisms of the torus $T^n$. A diffeomorphism is c.f. if every smooth function $f$ on $T^n$ is cohomologous to a constant $f_0$, i.e. there exists a $C^{\infty}$ function $h$ so that $h-h\circ\varphi=f-f_0$. We show that the only c.f. diffeomorphisms of $T^n$, $1\le n\le3$, are the smooth conjugations of Diophantine translations. For $n=4$, we prove the same result for c.f. orientation-preserving diffeomorphisms.

In this paper we prove the existence of an entire transcendental function of arbitrarily slow growth which exhibits simple dynamics on its Fatou set.

It is shown that under certain conditions a hyperbolic set will possess an invariant structure in addition to the invariant submanifolds. This invariant structure is in the form of a ‘jet of a foliation’ along the leaves.

Makienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.