In this paper we develop a generalization of foliated manifolds in the context of metric spaces. In particular we study dendritations of surfaces that are defined as maximal atlases of compatible upper semicontinuous local decompositions into dendrites. Applications are given in modeling stable and unstable sets of topological dynamical systems. For this purpose new forms of expansivity are defined.

We derive two fixed point theorems for a class of metric spaces that includes all Banach spaces and all complete Busemann spaces. We obtain our results by the use of a $1$-Lipschitz barycenter construction and an existence result for invariant Radon probability measures. Furthermore, we construct a bounded complete Busemann space that admits an isometry without fixed points.

For non-uniformly expanding maps inducing with a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining higher-order asymptotics for the correlation function in the infinite measure setting. Along the way, we show that these conditions are sufficient to recover previous results on sharp mixing rates in the finite measure setting for non-Markov maps, but for a larger class of observables. The results are illustrated by (finite and infinite measure-preserving) non-Markov interval maps with an indifferent fixed point.

Ergodic and combinatorial results obtained in Bergelson and Moreira [Ergodic theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns. Ergod. Th. & Dynam. Sys. to appear, published online 6 October 2015, doi:10.1017/etds.2015.68], involved measure preserving actions of the affine group of a countable field $K$. In this paper, we develop a new approach, based on ultrafilter limits, which allows one to refine and extend the results obtained in Bergelson and Moreira, op. cit., to a more general situation involving measure preserving actions of the non-amenable affine semigroups of a large class of integral domains. (The results and methods in Bergelson and Moreira, op. cit., heavily depend on the amenability of the affine group of a field.) Among other things, we obtain, as a corollary of an ultrafilter ergodic theorem, the following result. Let $K$ be a number field and let ${\mathcal{O}}_{K}$ be the ring of integers of $K$. For any finite partition $K=C_{1}\cup \cdots \cup C_{r}$, there exists $i\in \{1,\ldots ,r\}$ such that, for many $x\in K$ and many $y\in {\mathcal{O}}_{K}$, $\{x+y,xy\}\subset C_{i}$.

In the context of stationary $\mathbb{Z}^{d}$ nearest-neighbour Gibbs measures $\unicode[STIX]{x1D707}$ satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property) on the support of $\unicode[STIX]{x1D707}$ that is sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of topological strong spatial mixing for studying strong spatial mixing on systems with hard constraints. We also show that topological strong spatial mixing is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by Gamarnik and Katz [Sequential cavity method for computing free energy and surface pressure. J. Stat. Phys.137(2) (2009), 205–232], and Marcus and Pavlov [An integral representation for topological pressure in terms of conditional probabilities. Israel J. Math.207(1) (2015), 395–433], who gave a special representation of topological pressure in terms of conditional probabilities.

We strengthen and extend in this paper some recent results by Di Nasso, Goldbring, Jin, Leth, Lupini and Mahlburg on piecewise syndeticity of product sets in countable amenable groups to general countable measured groups. We also address several fundamental differences between the behavior of products of ‘large’ sets in Liouville and non-Liouville measured groups. As a (very) special case of our main results, we show that if $G$ is a free group of finite rank, and $A$ and $B$ are ‘spherically large’ subsets of $G$, then there exists a finite set $F\subset G$ such that $AFB$ is thick. The position of the set $F$ is curious, but seems to be necessary; in fact, we can produce left thick sets $A,B\subset G$ such that $B$ is ‘spherically large’, but $AB$ is not piecewise syndetic. On the other hand, if $A$ is spherically large, then $AA^{-1}$ is always piecewise syndetic and piecewise left syndetic. However, contrary to what happens for amenable groups, $AA^{-1}$ may fail to be syndetic. The same phenomena occur for many other (even amenable, but non-Liouville) measured groups. Our proofs are based on some ergodic-theoretical results concerning stationary actions which should be of independent interest.

We prove that for any set $E\subseteq \mathbb{Z}$ with upper Banach density $d^{\ast }(E)>0$, the set ‘of cubic configurations’ in $E$ is large in the following sense: for any $k\in \mathbb{N}$ and any $\unicode[STIX]{x1D700}>0$, the set

$$\begin{eqnarray}\displaystyle \biggl\{(n_{1},\ldots ,n_{k})\in \mathbb{Z}^{k}:d^{\ast }\biggl(\mathop{\bigcap }_{e_{1},\ldots ,e_{k}\in \{0,1\}}(E-(e_{1}n_{1}+\cdots +e_{k}n_{k}))\biggr)>d^{\ast }(E)^{2^{k}}-\unicode[STIX]{x1D700}\biggr\} & & \displaystyle \nonumber\end{eqnarray}$$

$\text{AVIP}_{0}^{\ast }$

$e_{1}p_{1}(n)+\cdots +e_{k}p_{k}(n)$

$p_{i}:\mathbb{Z}^{d}\longrightarrow \mathbb{Z}$

It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence and hence for tameness. In this paper, on any $3$-manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed $3$-manifold $M$, the space $\text{Diff}^{1}(M)$ admits a non-empty open set where every $C^{1}$-generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced in the authors’ previous paper. In order to eject the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partitions, which we introduce in this paper.

We consider stationary stochastic processes $\{X_{n}:n\in \mathbb{Z}\}$ such that $X_{0}$ lies in the closed linear span of $\{X_{n}:n\neq 0\}$; following Ghosh and Peres, we call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be linearly rigid, that the spectral density vanishes at zero and belongs to the Zygmund class $\unicode[STIX]{x1D6EC}_{\ast }(1)$. We next give a sufficient condition for stationary determinantal point processes on $\mathbb{Z}$ and on $\mathbb{R}$ to be linearly rigid. Finally, we show that the determinantal point process on $\mathbb{R}^{2}$ induced by a tensor square of Dyson sine kernels is not linearly rigid.

We are interested in attracting sets of $\mathbb{P}^{k}(\mathbb{C})$ which are of small topological degree and of codimension one. We first show that there exists a large family of examples. Then we study their ergodic and pluripotential theoretic properties.

For every $p>2$, we construct a regular and continuous specification ($g$-function), which has a variation sequence that is in $\ell ^{p}$ and which admits multiple Gibbs measures. Combined with a result of Johansson and Öberg [Square summability of variations of $g$-functions and uniqueness in $g$-measures. Math. Res. Lett.10(5–6) (2003), 587–601], this determines the optimal modulus of continuity for a specification which admits multiple Gibbs measures.

We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley–Zehnder index of a fixed point changes. The main result is that when the Conley–Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.

We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

Two notions of metric invariance entropy are constructed with respect to conditionally invariant measures for control systems in discrete time and it is shown that they are invariant under conjugacies.

We prove that when $f$ is a continuous self-map acting on a compact metric space $(X,d)$ that satisfies the shadowing property, then the set of irregular points (i.e., points with divergent Birkhoff averages) has full entropy. Using this fact, we prove that, in the class of $C^{0}$-generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbits of such points converges to some Sinai–Ruelle–Bowen-like measure in the weak$^{\ast }$ topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.

We give examples of locally constant $\text{SL}(2,\mathbb{R})$-cocycles over a Bernoulli shift that are discontinuity points for Lyapunov exponents in the Hölder topology and are arbitrarily close to satisfying the fiber bunching inequality. Backes, Brown, and the author [Continuity of Lyapunov exponents for cocycles with invariant holonomies. Preprint, 2015, arXiv:1507.08978] have shown that the Lyapunov exponents vary continuously when restricted to the space of fiber bunched Hölder continuous cocycles. Our examples give evidence that this theorem is optimal within certain families of Hölder cocycles.

We analyze the structure of the quotient $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We introduce a variant of an abstract construction of Fritz which encapsulates the convex combination operation on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$. This formalism allows us to define the geometric notion of an extreme point. We also discuss a topology on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ due to Abért and Elek in which it is Polish and compact, and show that this topology is equivalent to others defined in the literature. We show that the convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is compatible with the topology, and as a consequence deduce that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is path connected. Using ideas of Tucker-Drob, we are able to give a complete description of the topological and convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ for amenable $\unicode[STIX]{x1D6E4}$ by identifying it with the simplex of invariant random subgroups. In particular, we conclude that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ can be represented as a compact convex subset of a Banach space if and only if $\unicode[STIX]{x1D6E4}$ is amenable. In the case of general $\unicode[STIX]{x1D6E4}$ we prove a Krein–Milman-type theorem asserting that finite convex combinations of the extreme points of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ are dense in this space. We also consider the space $\text{A}_{{\sim}_{s}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of stable weak equivalence classes and show that it can always be represented as a compact convex subset of a Banach space. In the case of a free group $\mathbb{F}_{N}$, we show that if one restricts to the compact convex set $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})\subseteq \text{A}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$ consisting of the stable weak equivalence classes of free actions, then the extreme points are dense in $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$.

We study a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds, that is, a decorated graph. We consider the following dynamical system on decorated graphs. Suppose that, at the initial time, we have a narrow wave packet on a one-dimensional edge. It can be thought of as a point moving along the edge. When a packet arrives at the point of gluing, the expanding wavefront begins to spread on the Riemannian manifold. At the same time, there is a partial reflection of the wave packet. When the wavefront that propagates on the surface comes to another point of gluing, it generates a reflected wavefront and a wave packet on an edge. We study the number of Gaussian packets, that is, moving points on one-dimensional edges as time goes to infinity. We prove the asymptotic estimations for this number for the following decorated graphs: a cylinder with an interval, a two-dimensional torus with an interval and a three-dimensional torus with an interval. Also we prove a general theorem about a manifold with an interval and apply it to the case of a uniformly secure manifold.

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the average of this sequence times any nilsequence converges. Two decomposition results of independent interest play key roles in the proof. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has small uniformity norm (this generalizes a result of the second author and Kra); and the second states that every multiple correlation sequence in several variables is a sum of a nilsequence and a sequence that is small in uniform density (this generalizes a result of the first author). Furthermore, we use these results in order to establish mean convergence and recurrence results for a variety of sequences of dynamical and arithmetic origin and give some combinatorial implications.

For a self-similar or self-affine iterated function system (IFS), let $\unicode[STIX]{x1D707}$ be the self-similar or self-affine measure and $K$ be the self-similar or self-affine set. Assume that the IFS satisfies the weak separation condition and $K$ is totally disconnected; then, by using the technique of neighborhood decomposition, we prove that there is a neighborhood $\unicode[STIX]{x1D6FA}$ of the identity map Id such that $\sup \{\unicode[STIX]{x1D707}(g(K)\cap K):g\in \unicode[STIX]{x1D6FA}\setminus \{\text{Id}\}\}<1$.