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.

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 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

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

We compute the Poisson boundary of locally discrete groups of diffeomorphisms of the circle.

In order to understand the parameter space Ξd of monic and centered complex polynomial vector fields in ℂ of degree d, decomposed by the combinatorial classes of the vector fields, it is interesting to know the number of loci in parameter space consisting of vector fields with the same combinatorial data (corresponding to topological classification with fixed separatrices at infinity). This paper answers questions posed by Adam L. Epstein and Tan Lei about the total number of combinatorial classes and the number of combinatorial classes corresponding to loci of a specific (real) dimension q in parameter space, for fixed degree d; these numbers are denoted by cd and cd,q, respectively. These results are extensions of a result by Douady, Estrada, and Sentenac, which shows that the number of combinatorial classes of the structurally stable complex polynomial vector fields in ℂ of degree d is the Catalan number Cd−1. We show that enumerating the combinatorial classes is equivalent to a so-called bracketing problem. Then we analyze the generating functions and find closed-form expressions for cd and cd,q, and we furthermore make an asymptotic analysis of these sequences for d tending to ∞. These results are also applicable to special classes of quadratic and Abelian differentials and singular holomorphic foliations of the plane.

We give a complete characterization of the compact metric dynamical systems that appear as boundaries of the canonical compactification of a locally compact countable state mixing Markov shift. Consider such a compact metric dynamical system. Then there is a pair of non-conjugate Markov shifts with conjugate canonical compactifications, one of which has the given compact system as canonical boundary.

If (M,F) is a C4 compact Finsler surface of genus at least two without conjugate points, we show that the first integrals of the geodesic flow are constant. Using this fact, we show that if (M,F) is also of Landsberg type then (M,F) is Riemannian. The connection between the absence of conjugate points and the Riemannian character of the Finsler metric has some remarkable consequences concerning rigidity.

Let (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then μ({x∈X:x∈Bi infinitely often})=1. Suppose (T,X,μ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether Tix∈Bi infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If Tix∈Bi infinitely often for μ almost every x, we call the sequence (Bi) a Borel–Cantelli sequence. If the sets Bi :=B(p,ri) are nested balls of radius ri about a point p, then the question of whether Tix∈Bi infinitely often for μ almost every x is often called the shrinking target problem. We show, under certain assumptions on the measure μ, that for balls Bi if μ(Bi)≥i−γ, 0<γ<1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies that the sequence is Borel–Cantelli. If μ(Bi)≥C1 /i, then exponential decay of correlations implies that the sequence is Borel–Cantelli. We give conditions in terms of return time statistics which quantify Borel–Cantelli results for sequences of balls such that μ(Bi)≥C/i. Corollaries of our results are that for planar dispersing billiards and Lozi maps, sequences of nested balls B(p,1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.

We present amalgamated free products satisfying coupling rigidity with respect to the automorphism group of the associated Bass–Serre tree. As an application, we obtain orbit equivalence rigidity for amalgamated free products of mapping class groups.

In this paper we compute the derivative of the action on probability measures of an expanding circle map at its absolutely continuous invariant measure. The derivative is defined using optimal transport: we use the rigorous framework set up by Gigli to endow the space of measures with a kind of differential structure. It turns out that 1 is an eigenvalue of infinite multiplicity of this derivative, and we deduce that the absolutely continuous invariant measure can be deformed in many ways into atomless, nearly invariant measures. We also show that the action of standard self-covering maps on measures has positive metric mean dimension.

We prove that every non-singular Bernoulli shift is either zero-type or there is an equivalent invariant stationary product probability. We also give examples of a type III1Bernoulli shift and a Markovian flow which are power weakly mixing and zero-type.

For every ergodic hyperbolic measure ω of a C1+α diffeomorphism, there is an ω-full-measure set $\tilde {\Lambda }$ (the union of $\tilde \Lambda _l=\mathrm {supp}( \omega |_{\Lambda _{l}})$, the support sets of ω on each Pesin block Λl, l=1,2,…) such that every non-empty, compact and connected subset $V\subseteq \mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ coincides with Vf(x), where $\mathcal {M}_{\mathrm {inv}}(\tilde {\Lambda })$ denotes the space of invariant measures supported on $\tilde {\Lambda }$ and Vf(x) denotes the accumulation set of time averages of Dirac measures supported at one orbit of some point x. For each fixed set V, the points with the above property are dense in the support supp (ω) . In particular, points satisfying $V_f(x)=\mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ are dense in supp (ω) . Moreover, if supp (ω) is isolated, the points satisfying $V_f(x)\supseteq \mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ form a residual subset of supp (ω) . These extend results of K. Sigmund [On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285–299] (see also M. Denker, C. Grillenberger and K. Sigmund [Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, Berlin, Ch. 21]) from the uniformly hyperbolic case to the non-uniformly hyperbolic case. As a corollary, irregular + points form a residual set of supp (ω) .

In 1995, Hill and Velani introduced the ‘shrinking targets’ theory. Given a dynamical system ([0,1],T), they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well approximated by orbits {Tnx}n≥0, where Tis an expanding Markov map with a finite partition supported by [0,1]. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.

We give a general version of the Birkhoff ergodic theorem for functions taking values in non-positively curved spaces. In this setting, the notion of a Birkhoff sum is replaced by that of a barycenter along the orbits. The construction of an appropriate barycenter map is the core of this note. As a byproduct of our construction, we prove a fixed point theorem for actions by isometries on a Buseman space.

We investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on ℝd. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results ‘partially’ answer one of the fundamental questions posed in [13]—to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e. they have unbounded variation.