On the space of signed invariant measures of Aℕ, one constructs a norm (and hence a distance) that seems to have a particular significance in dynamics. I shall present some of its properties, in particular a duality theorem à la Kantorovich–Rubinshtein, which gives an expression of this distance using couplings.

We prove a theorem on the structural stability of smooth attractor–repellor endomorphisms of compact manifolds, with singularities. By attractor–repellor, we mean that the non-wandering set of the dynamics f is the disjoint union of an expanding compact subset with a hyperbolic attractor on which f acts bijectively. The statement of this result is both infinitesimal and dynamical. To our knowledge, this is the first in this hybrid direction. Our results also generalize Mather’s theorem in singularity theory, which states that infinitesimal stability implies structural stability for composed mappings to the larger category of laminations.

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

Let f be an arbitrary transcendental entire or meromorphic function in the class 𝒮 (i.e. with finitely many singularities). We show that the topological pressure P(f,t) for t>0 can be defined as the common value of the pressures P(f,t,z) for all z∈ℂ up to a set of Hausdorff dimension zero. Moreover, we prove that P(f,t) equals the supremum of the pressures of f∣X over all invariant hyperbolic subsets X of the Julia set, and we prove Bowen’s formula for f, i.e. we show that the Hausdorff dimension of the radial Julia set of f is equal to the infimum of the set of t, for which P(f,t) is non-positive. Similar results hold for (non-exceptional) transcendental entire or meromorphic functions f in the class ℬ (i.e. with a bounded set of singularities), for which the closure of the post-singular set does not contain the Julia set.

Let Σ be a finite alphabet, Ω=Σℤd equipped with the shift action, and ℐ the simplex of shift-invariant measures on Ω. We study the relation between the restriction ℐn of ℐ to the finite cubes {−n,…,n}d⊂ℤd, and the polytope of ‘locally invariant’ measures ℐlocn. We are especially interested in the geometry of the convex set ℐn, which turns out to be strikingly different when d=1 and when d≥2 . A major role is played by shifts of finite type which are naturally identified with faces of ℐn, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of ℐn, although in dimension d≥2 there are also extreme points which arise in other ways. We show that ℐn =ℐlocn when d=1 , but in higher dimensions they differ for n large enough. We also show that while in dimension one ℐn are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of ℐn for all large enough n.

We show that certain billiard tables with non-compact cusps are mixing with respect to the invariant infinite measure, in the sense of Krengel and Sucheston. Moreover, we show that the scaling rate is slower than a certain polynomial rate.

We prove the existence of a topologically mixing interval exchange transformation and prove that no interval exchange is topologically mixing of all orders.

In the last two decades, considerable attention has been paid to the dimension theory of self-affine sets. In the case of generalized 4-corner sets (see Figure 1), the iterated function systems obtained as the projections of self-affine systems have maps of common fixed points. In this paper, we extend our result [B. Bárány. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fund. Math. 206 (2009), 49–59], which introduced a new method of computation of the box and Hausdorff dimensions of self-similar families where some of the maps have common fixed points. The extended version of our method presented in this paper makes it possible to determine the box dimension of the generalized 4-corner set for Lebesgue-typical contracting parameters.

We study the Toeplitz algebra 𝒯(ℕ⋊ℕ×) and three quotients of this algebra: the C*-algebra 𝒬ℕ recently introduced by Cuntz, and two new ones, which we call the additive and multiplicative boundary quotients. These quotients are universal for Nica-covariant representations of ℕ⋊ℕ× satisfying extra relations, and can be realised as partial crossed products. We use the structure theory for partial crossed products to prove a uniqueness theorem for the additive boundary quotient, and use the recent analysis of KMS states on 𝒯(ℕ⋊ℕ×) to describe the KMS states on the two quotients. We then show that 𝒯(ℕ⋊ℕ×), 𝒬ℕ and our new quotients are all interesting new examples for Larsen’s theory of Exel crossed products by semigroups.

We show that every uniquely ergodic minimal Cantor system is topologically orbit equivalent to the natural extension of a numeration scale associated to a logistic map.

The editors are reproducing here an informal research statement that Dan wrote in one of his last years at the University of Maryland. It provides a rare insight into how he viewed his own work and as usual he did an excellent job in setting forth the main points in a clear way. The summary was retyped, lightly edited to correct typos, and provided with references by Ayşe Şahin. One of the editors (B.W.) added a brief appendix to cover some of the highlights of Dan’s work during the last years of his life. Following the research summary is a complete list of Dan’s publications as well as a second list for the other references cited.

We study the limiting behavior of multiple ergodic averages involving several, not necessarily commuting, measure-preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and another that uses iterates along shifted polynomials. We prove pointwise convergence in both cases, thus answering a question of I. Assani in the former case, and extending the results of B. Host and B. Kra, and A. Leibman in the latter case. Our argument is based on some elementary uniformity estimates of general bounded sequences, decomposition results in ergodic theory, and equidistribution results on nilmanifolds.

We use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. In previous work, we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.

We show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mañé example and one obtained through a Hopf bifurcation.

Let α1,…,αn be irrational numbers which are linearly independent over the rationals, and f1,…,fn commuting germs of holomorphic diffeomorphisms in ℂ such that fk(0)=0,f′k(0)=e2πiαk,k=1,…,n. Moser showed that f 1,…,fn are simultaneously linearizable (i.e. conjugate by a germ of holomorphic diffeomorphism h to the rigid rotations Rαk(z)=e2πiαkz) if the vector of rotation numbers (α 1,…,αn) satisfies a Diophantine condition. Adapting Yoccoz’s renormalization to the setting of commuting germs, we show that simultaneous linearization holds in the presence of a weaker Brjuno-type condition ℬ(α1,…,αn)<+∞, where ℬ(α1,…,αn) is a multivariable analogue of the Brjuno function. If there are no periodic orbits for the action of the germs f1,…,fn in a neighbourhood of the origin, then a weaker arithmetic condition ℬ′ (α 1,…,αn)<+∞ analogous to Perez-Marco’s condition for linearization in the absence of periodic orbits is shown to suffice for linearizability. Normalizing the germs to be univalent on the unit disc, in both cases the Siegel discs are shown to contain discs of radii for some universal constant C.

We define the action of a locally compact group G on a topological graph E. This action induces a natural action of G on the C*-correspondence ℋ(E) and on the graph C*-algebra C*(E). If the action is free and proper, we prove that C*(E)⋊rG is strongly Morita equivalent to C*(E/G) . We define the skew product of a locally compact group G by a topological graph E via a cocycle c:E1 →G. The group acts freely and properly on this new topological graph E×cG. If G is abelian, there is a dual action on C* (E) such that . We also define the fundamental group and the universal covering of a topological graph.

This paper is an update and extension of a result the authors first proved in 2003. The goal of this paper is to study factors which are known to be L2-characteristic for certain non-conventional averages and prove that these factors are pointwise characteristic for the multiterm return times averages.

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0–1 sequences (xk) such that xkx2k=0 for all k. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for multiplicative subshifts.