Let $C$ be a locally convex closed subset of a negatively curved Riemannian manifold $M$. We define the skinning measure ${\sigma }_{C} $ on the outer unit normal bundle to $C$ in $M$ by pulling back the Patterson–Sullivan measures at infinity, and give a finiteness result for ${\sigma }_{C} $, generalizing the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to $C$ equidistribute to the Bowen–Margulis measure ${m}_{\mathrm{BM} } $ on ${T}^{1} M$, assuming only that ${m}_{\mathrm{BM} } $ is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.

We present new examples of open sets of diffeomorphisms such that generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that all other classes except one are contained in periodic surfaces. This study allows us to obtain the existence of Milnor attractors as well as study ergodic properties of the diffeomorphisms in those open sets by using ideas and results from Bonatti and Viana [SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting. Israel J. Math.115 (2000), 157–193] and Buzzi and Fisher [Entropic stability beyond partial hyperbolicity. Preprint, 2011, arXiv:1103:2707].

Let the circle act symplectically on a compact, connected symplectic manifold $M$. If there are exactly three fixed points, $M$ is equivariantly symplectomorphic to $\mathbb{CP}^{2}$.

The non-existence of a real meromorphic first integral for a spherically confined steady Stokes flow of Bajer and Moffatt is proved on the basis of Ziglin’s theory and the differential Galois theory. In the proof, the differential Galois group of a second-order Fuchsian-type differential equation associated with normal variations along a particular streamline is shown to be a special linear group according to Kovacic’s algorithm. A set of special values of a parameter contained in the Fuchsian-type equation is studied by using the theory of elliptic curves. For this set, a computer algebra system is used in part of Kovacic’s algorithm.

We give a unified treatment of decay of correlations for non-uniformly expanding systems with a good inducing scheme. In addition to being more elementary than previous treatments, our results hold for general integrable return time functions under fairly mild conditions on the inducing scheme.

Inspired by topological Wiener–Wintner theorems we study the mean ergodicity of amenable semigroups of Markov operators on $C(K)$ and show the connection to the convergence of strong and weak ergodic nets. The results are then used to characterize mean ergodicity of Koopman semigroups corresponding to skew product actions on compact group extensions.

We introduce a quantitative condition on orbits of dynamical systems, which measures their aperiodicity. We show the existence of sequences in the Bernoulli shift and geodesics on closed hyperbolic manifolds which are as aperiodic as possible with respect to this condition.

Nathanial Brown [Topological dynamical systems associated to ${\mathit{II}}_{1} $-factors. Adv. Math. 227(4), 1665–1699] introduced a convex-like structure on the set of unitary equivalence classes of unital *-homomorphisms of a separable type ${\mathit{II}}_{1} $ factor into ${R}^{\omega } $ (ultrapower of the hyperfinite factor). The goal of this paper is to introduce such a structure on the set of sofic representations of groups. We prove that if the commutant of a representation acts ergodically on the Loeb measure space then that representation is an extreme point.

We study a measure entropy for finitely generated free group actions called f-invariant entropy. The f-invariant entropy was developed by L. Bowen and is essentially a special case of his measure entropy theory for actions of sofic groups. In this paper we relate the f-invariant entropy of a finitely generated free group action to the f-invariant entropy of the restricted action of a subgroup. We show that the ratio of these entropies equals the index of the subgroup. This generalizes a well-known formula for the Kolmogorov–Sinai entropy of amenable group actions. We then extend the definition of f-invariant entropy to actions of finitely generated virtually free groups. We also obtain a numerical virtual measure conjugacy invariant for actions of finitely generated virtually free groups.

We study hyperbolized versions of cohomological equations that appear with cocycles by isometries of the Euclidean space. These (hyperbolized versions of) equations have a unique continuous solution. We concentrate on the question whether or not these solutions converge to a genuine solution to the original equation, and in what sense we can use them as good approximative solutions. The main advantage of considering solutions to hyperbolized cohomological equations is that they can be easily described, since they are global attractors of a naturally defined skew-product dynamics. We also include some technical results about twisted Birkhoff sums and exponential averaging.

We study the group Diffr0(𝔸) of Cr-diffeomorphisms of the closed annulus that are isotopic to the identity. We show that, for r≠2,3, the linear space of homogeneous quasi-morphisms on the group Diffr0(𝔸) is one-dimensional. Therefore, the commutator length on this group is (stably) unbounded. In particular, this provides an example of a manifold whose diffeomorphism group is unbounded in the sense of Burago, Ivanov and Polterovich.

We completely classify the extreme tracial states on the cores of the ${C}^{\ast } $-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.

Roth type irrational rotation numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this paper we investigate how to generalize Roth-like Diophantine conditions to interval exchange maps. If one considers the dynamics in parameter space one can introduce two non-equivalent Roth type conditions, the first (condition (Z)) by means of the Zorich cocycle [Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier 46(2) (1996), 325–370], the second (condition (A)) by means of a further acceleration of the continued fraction algorithm by Marmi–Moussa–Yoccoz introduced in [The cohomological equation for Roth type interval exchange maps, J. Amer. Math. Soc. 18 (2005), 823–872]. A third very natural condition (condition (D)) arises by considering the distance between the discontinuity points of the iterates of the map. If one considers the dynamics of an interval exchange map in phase space then one can introduce the notion of Diophantine type by considering the asymptotic scaling of return times pointwise or with respect to uniform convergence (respectively conditions (R) and (U)). In the case of circle rotations all the above conditions are equivalent. For interval exchange maps of three intervals we show that (D) and (A) are equivalent and imply (Z), (U) and (R), which are equivalent among them. For maps of four intervals or more we prove several results; the only relation that we cannot decide is whether (Z) implies (R) or not.

Linked-twist maps are area-preserving, piecewise diffeomorphisms, defined on a subset of the torus. They are non-uniformly hyperbolic generalizations of the well-known Arnold cat map. We show that a class of canonical examples have polynomial decay of correlations for $\alpha $-Hölder observables, of order $1/ n$.

We prove homotopical rigidity of expanding dynamical systems, by showing that they are determined by a group-theoretic invariant. We use this to show that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules. Moreover, the cut-and-paste rules can be found algorithmically from the algebraic invariant.

We study the invariant distributions for horocycle maps on $\Gamma \backslash SL(2, \mathbb {R})$and prove Sobolev estimates for the cohomological equation of horocycle maps. As an application, we obtain a rate of equidistribution for horocycle maps on compact manifolds.

Let $G$ be a countable abelian group and let ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ be measure preserving $G$-actions on a probability space. We prove that joint ergodicity of ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ implies total joint ergodicity if each ${T}^{(i)} $ is totally ergodic. We also show that if $G= { \mathbb{Z} }^{d} $, $s\geq d+ 1$ and the actions ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ commute, then total joint ergodicity of ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ follows from joint ergodicity. This can be seen as a generalization of Berend’s result for commuting $ \mathbb{Z} $-actions.

Let φ:ℙN⤏ℙN be a dominant rational map. The dynamical degree of φ is the quantity δφ=lim (deg φn)1/n. When φ is defined over ${\bar {{\mathbb {Q}}}}$, we define the arithmetic degree of a point $P\in {\mathbb {P}}^N({\bar {{\mathbb {Q}}}})$ to be αφ(P)=lim sup h(φn(P))1/n and the canonical height of P to be $\hat {h}_\varphi (P)=\limsup \delta _\varphi ^{-n}n^{-\ell _\varphi }h(\varphi ^n(P))$ for an appropriately chosen ℓφ. We begin by proving some elementary relations and making some deep conjectures relating δφ, αφ(P) , ${\hat h}_\varphi (P)$, and the Zariski density of the orbit 𝒪φ(P) of P. We then prove our conjectures for monomial maps.

We give a new proof of Rudolph’s multiple term return times theorem based on Host–Kra structure theory. Our approach provides characteristic factors for all terms, works for arbitrary tempered Følner sequences and also yields a multiple term Wiener–Wintner-type return times theorem for nilsequences.

In this paper, we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new example in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial computable functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattices ℰs and ℰw of Medvedev degrees and Muchnik degrees of non-empty effectively closed subsets of {0,1}ℕ. We now prove that ℰs and ℰwconsist precisely of the Medvedev degrees and Muchnik degrees of two-dimensional subshifts of finite type. We apply this result to obtain an infinite collection of two-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.