Let Γ be a countable group and let f be a free probability measure-preserving action of Γ. We show that all Bernoulli actions of Γ are weakly contained in f. It follows that for a finitely generated group Γ, the cost is maximal on Bernoulli actions for Γ and that all free factors of i.i.d. (independent and identically distributed) actions of Γ have the same cost. We also show that if f is ergodic, but not strongly ergodic, then f is weakly equivalent to f×I, where Idenotes the trivial action of Γ on the unit interval. This leads to a relative version of the Glasner–Weiss dichotomy.

Rational weak mixing is a measure theoretic version of Krickeberg’s strong ratio mixing property for infinite measure preserving transformations. It requires ‘density’ ratio convergence for every pair of measurable sets in a dense hereditary ring. Rational weak mixing implies weak rational ergodicity and (spectral) weak mixing. It is enjoyed for example by Markov shifts with Orey’s strong ratio limit property. The power, subsequence version of the property is generic.

We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araújo [Random perturbations of non-uniformly expanding maps. Astérisque 286 (2003), 25–62], where the stochastic stability in the $\mathrm {weak}^*$ topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai–Ruelle–Bowen (SRB) measure of the unperturbed system in the $L^1$-norm. As an application of our results, we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in Alves, Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398] and the second one is the class of Viana maps.

We prove that for all ergodic extensions $S_{1}$ of a transformation by a locally compact second countable group $G$, and for all $G$-extensions $ S_{2} $ of an aperiodic transformation, there is a relative speedup of $ S_{1} $ that is relatively isomorphic to $S_{2}$. We apply this result to give necessary and sufficient conditions for two ergodic $n$-point or countable extensions to be related in this way.

Many results of the Fatou–Julia iteration theory of rational functions extend to uniformly quasiregular maps in higher dimensions. We obtain results of this type for certain classes of quasiregular maps which are not uniformly quasiregular.

Given positive integers $n$ and $m$, we consider dynamical systems in which (the disjoint union of) $n$ copies of a topological space is homeomorphic to $m$ copies of that same space. The universal such system is shown to arise naturally from the study of a C*-algebra denoted by ${\cal O}_{m,n}$, which in turn is obtained as a quotient of the well-known Leavitt C*-algebra $L_{m,n}$, a process meant to transform the generating set of partial isometries of $L_{m,n}$ into a tame set. Describing ${\cal O}_{m,n}$ as the crossed product of the universal $(m,n)$-dynamical system by a partial action of the free group $\mathbb {F}_{m+n}$, we show that ${\cal O}_{m,n}$ is not exact when $n$ and $m$ are both greater than or equal to 2, but the corresponding reduced crossed product, denoted by ${\cal O}_{m,n}^r$, is shown to be exact and non-nuclear. Still under the assumption that $m,n\geq 2$, we prove that the partial action of $\mathbb {F}_{m+n}$ is topologically free and that ${\cal O}_{m,n}^r$ satisfies property (SP) (small projections). We also show that ${\cal O}_{m,n}^r$admits no finite-dimensional representations. The techniques developed to treat this system include several new results pertaining to the theory of Fell bundles over discrete groups.

Extending a result of Mañé, Hayashi proved that every diffeomorphism $f$ which has a $C^1$-neighborhood $\mathcal {U}$ where all periodic points of any $g\in \mathcal {U}$are hyperbolic is an Axiom A diffeomorphism. Here we prove the analogous result in the volume-preserving scenario.

Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations.

We consider locally minimizing measures for conservative twist maps of the $d$-dimensional annulus and for Tonelli Hamiltonian flows defined on a cotangent bundle $T^*M$. For weakly hyperbolic measures of such type (i.e. measures with no zero Lyapunov exponents), we prove that the mean distance/angle between the stable and unstable Oseledets bundles gives an upper bound on the sum of the positive Lyapunov exponents and a lower bound on the smallest positive Lyapunov exponent. We also prove some more precise results.

The structure of a self-similar set with the open set condition does not change under magnification. For self-affine sets, the situation is completely different. We consider self-affine Cantor sets $E\subset \mathbb {R}^2$ of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that in small square $\varepsilon $-neighborhoods $N$ of almost each point $x$ in $E,$ with respect to many Bernoulli measures on the address space, $E\cap N$ is well approximated by product sets $[0,1]\times C$, where $C$ is a Cantor set. Even though $E$ is totally disconnected, all tangent sets have a product structure with interval fibers, reminiscent of the view of attractors of chaotic differentiable dynamical systems. We also prove that $E$has uniformly scaling scenery in the sense of Furstenberg, Gavish and Hochman: the family of tangent sets is the same at almost all points$x.$

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.

Consider an irrational rotation of a unit circle and a real continuous function. A point is declared ‘maximizing’ if the growth of the ergodic sums at this point is maximal up to an additive constant. In the case of two-sided ergodic sums, the existence of a maximizing point for a continuous function implies that it is the coboundary of a continuous function. In contrast, we build, for the ‘usual’ one-sided ergodic sums, examples in Hölder or smooth classes, indicating that all kinds of behaviour of the function with respect to the dynamical system are possible. We also show that generic continuous functions are without maximizing points, not only for rotations, but also for the transformation 2x mod 1. For this latter transformation, it is known that any Hölder continuous function has a maximizing point.

We show that for a large class of piecewise expanding maps T, the bounded p-variation observables u0 that admit an infinite sequence of bounded p-variation observables ui satisfying

\[ u_{i}= u_{i+1}\circ T-u_{i+1} \]

Suppose $\sigma $ is the shift acting on Bernoulli space $X=\{0,1\}^{\mathbb {N}}$, and consider a fixed function $f:X \to \mathbb {R}$ satisfying the Walters conditions (defined in [P. Walters. A natural space of functions for the Ruelle operator theorem. Ergod. Th. & Dynam. Sys.27 (2007), 1323–1348]). For each real value $t\geq 0$ we consider the Ruelle operator $L_{\mathit {tf}}$. We are interested in the main eigenfunction $h_t$ of $L_{\mathit {tf}}$ and the main eigenmeasure $\nu _t$ for the dual operator $L_{\mathit {tf}}^*$, which we consider normalized in such a way that $h_t(0^\infty )=1$ and $\int h_t \,d\nu _t=1$ for all $t\gt 0$. We denote by $\mu _t= h_t \nu _t$ the Gibbs state for the potential $\mathit {tf}$. By the selection of a subaction $V$, when the temperature goes to zero (or $t\to \infty $), we mean the existence of the limit

\[ V:=\lim _{t\to \infty }\frac {1}{t}\log (h_{t}). \]

$\mu $

$t\to \infty $

\[\mu :=\lim _{t\to \infty } \mu _t.\]

$f$

$f$

$V$

Thurston introduced $\sigma _d$-invariant laminations (where $\sigma _d(z)$ coincides with $z^d:\mathbb S ^1\to \mathbb S ^1$, $d\ge 2$) and defined wandering $k$-gons as sets ${\mathbf {T}}\subset \mathbb S ^1$ such that $\sigma _d^n({\mathbf {T}})$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\sigma _d^n({\mathbf {T}})$ in the plane are pairwise disjoint. He proved that $\sigma _2$ has no wandering $k$-gons. Call a lamination with wandering $k$-gons a WT-lamination. In a recent paper, it was shown that uncountably many cubic WT-laminations, with pairwise non-conjugate induced maps on the corresponding quotient spaces $J$, are realizable as cubic polynomials on their (locally connected) Julia sets. Here we use a new approach to construct cubic WT-laminations with the above properties so that any wandering branch point of $J$ has a dense orbit in each subarc of $J$ (we call such orbits condense), and show that critical portraits corresponding to such laminations are dense in the space ${\mathcal A}_3$of all cubic critical portraits.

We study the number of visits to balls Br(x), up to time t/μ(Br(x)), for a class of non-uniformly hyperbolic dynamical systems, where μ is the Sinai–Ruelle–Bowen measure. Outside a set of ‘bad’ centers x, we prove that this number is approximately Poissonnian with a controlled error term. In particular, when r→0, we get convergence to the Poisson law for a set of centers of μ-measure one. Our theorem applies for instance to the Hénon attractor and, more generally, to systems modelled by a Young tower whose return-time function has an exponential tail and with one-dimensional unstable manifolds. Along the way, we prove an abstract Poisson approximation result of independent interest.

Let $G$ be a connected nilpotent Lie group. Given probability-preserving$G$-actions $(X_i,\Sigma _i,\mu _i,u_i)$, $i=0,1,\ldots ,k$, and also polynomial maps $\phi _i:\mathbb {R}\to G$, $i=1,\ldots ,k$, we consider the trajectory of a joining $\lambda $ of the systems $(X_i,\Sigma _i,\mu _i,u_i)$ under the ‘off-diagonal’ flow

\[ (t,(x_0,x_1,x_2,\ldots ,x_k))\mapsto (x_0,u_1^{\phi _1(t)}x_1,u_2^{\phi _2(t)}x_2,\ldots ,u_k^{\phi _k(t)}x_k). \]

$\lambda $

$\lambda '$

$\lambda '$

$G^{k+1}$

In this paper we establish a Fréchet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying extreme value theory. Subsequently, we show that this law gives rise to an Erdős–Philipp law and to various generalized Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erdős.

Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a periodic orbit. Such perturbations preserve certain homoclinic relations and prescribed dynamical properties of a homoclinic class.

Given a transformation T of a standard measure space (X,μ), let ℳ(T) denote the set of spectral multiplicities of the Koopman operator UT defined in by UTf:=f∘T. In this survey paper we discuss which subsets of are realizable as ℳ(T) for various T: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure-preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed.