We establish a ‘diagonal’ ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg’s correspondence principle, prove that any ‘large’ set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular, we obtain an alternative proof for a result obtained by Cilleruelo [Combinatorial problems in finite fields and Sidon sets. Combinatorica32(5) (2012), 497–511], showing that for any finite field $F$ and any subsets $E_{1},E_{2}\subset F$ with $|E_{1}|\,|E_{2}|>6|F|$, there exist $u,v\in F$ such that $u+v\in E_{1}$ and $uv\in E_{2}$.

We will show that the sequence appearing in the double recurrence theorem is a good universal weight for the Furstenberg averages. That is, given a system $(X,{\mathcal{F}},\unicode[STIX]{x1D707},T)$ and bounded functions $f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$, there exists a set of full-measure $X_{f_{1},f_{2}}$ in $X$ that is independent of integers $a$ and $b$ and a positive integer $k$ such that, for all $x\in X_{f_{1},f_{2}}$, for every other measure-preserving system $(Y,{\mathcal{G}},\unicode[STIX]{x1D708},S)$ and for each bounded and measurable function $g_{1},\ldots ,g_{k}\in L^{\infty }(\unicode[STIX]{x1D708})$, the averages

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)g_{1}\circ S^{n}g_{2}\circ S^{2n}\cdots g_{k}\circ S^{kn}\end{eqnarray}$$

$L^{2}(\unicode[STIX]{x1D708})$

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

In this paper we give a formula for the $K$-theory of the $C^{\ast }$-algebra of a weakly left-resolving labelled space. This is done by realizing the $C^{\ast }$-algebra of a weakly left-resolving labelled space as the Cuntz–Pimsner algebra of a $C^{\ast }$-correspondence. As a corollary, we obtain a gauge-invariant uniqueness theorem for the $C^{\ast }$-algebra of any weakly left-resolving labelled space. In order to achieve this, we must modify the definition of the $C^{\ast }$-algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of $C^{\ast }$-algebras that are associated with shift spaces and labelled graph algebras. Hence, by computing the $K$-theory of a labelled graph algebra, we are providing a common framework for computing the $K$-theory of graph algebras, ultragraph algebras, Exel–Laca algebras, Matsumoto algebras and the $C^{\ast }$-algebras of Carlsen. We provide an inductive limit approach for computing the $K$-groups of an important class of labelled graph algebras, and give examples.

We study relations between transitivity, mixing and periodic points on dendrites. We prove that, when there is a point with dense orbit which is a cutpoint, periodic points are dense and there is a terminal periodic decomposition. We also show that it is possible that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It is also possible that a dynamical system is transitive but there is a unique periodic point which, in fact, is the unique fixed point. We also prove that on almost meshed continua (a class of continua containing topological graphs and dendrites with closed or countable set of endpoints), periodic points are dense if and only if they are dense for the map induced on the hyperspace of all non-empty compact subsets.

We consider a simple model of an open partially expanding map. Its trapped set ${\mathcal{K}}$ in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component $\unicode[STIX]{x1D708}$ in the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call ‘minimal captivity’. This hypothesis is stable under perturbations and means that the dynamics is univalued in a neighborhood of ${\mathcal{K}}$ . Under this hypothesis we show the existence of an asymptotic spectral gap and a fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit $\unicode[STIX]{x1D708}\rightarrow \infty$ . Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.

We discuss multiple versions of rational ergodicity and rational weak mixing for ‘nice’ transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.

We survey the theory of quasi-periodic Schrödinger-type operators, focusing on the advances made since the early 2000s by adopting a dynamical systems point of view.

We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set called the set of walls, which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relative to four orbits and describe them explicitly by adding a tangle to Handel’s diagram and to the set of walls. This is essentially an isotopy class of simple closed curves in the cylinder minus two points.

We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity. Also, we consider the latter phenomenon and characterize it among transformations admitting generalized recurrent events.

This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group $G$, Parry showed how to define a $G$-extension $S_{A}$ from a square matrix over $\mathbb{Z}_{+}G$, and classified the extensions up to topological conjugacy by the strong shift equivalence class of $A$ over $\mathbb{Z}_{+}G$. Parry asked, in this case, if the dynamical zeta function $\det (I-tA)^{-1}$ (which captures the ‘periodic data’ of the extension) would classify the extensions by $G$ of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic $\text{K}$-theory group $\text{NK}_{1}(\mathbb{Z}G)$ is non-trivial (e.g. for $G=\mathbb{Z}/n$ with $n$ not square-free) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of $\text{NK}_{1}(\mathbb{Z}G)$, for every non-trivial abelian $G$ we show that there exists a shift of finite type with an infinite family of mixing non-conjugate $G$ extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for $G$ (not necessarily abelian), and extend all the above results to the non-abelian case. There is other work on basic invariants. The constructions require the ‘positive $K$-theory’ setting for positive equivalence of matrices over $\mathbb{Z}G[t]$.

We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions as Poincaré maps of measured foliations, a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet, $A$ . The set of return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$ .

We provide an explicit lower bound for the the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model. In particular, for the Anderson model on a strip of width $W$, the lower bound is proportional to $W^{-\unicode[STIX]{x1D716}}$, for any $\unicode[STIX]{x1D716}>0$. This bound is consistent with the fact that the lowest non-negative Lyapunov exponent is conjectured to have a lower bound proportional to $W^{-1}$.

We extend some aspects of the smooth approximation by conjugation method to the real-analytic set-up, and create examples of zero entropy, uniquely ergodic, real-analytic diffeomorphisms of the two-dimensional torus that are metrically isomorphic to some (Liouvillian) irrational rotations of the circle.

In this paper we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if $X$ is a metric space, let $2^{X}$ denote the space of non-empty compact subsets of $X$ provided with the Hausdorff topology. If $f$ is a continuous self-map on $X$, there is a naturally induced continuous self-map $f_{\ast }$ on $2^{X}$. Our main theme is the interrelation between the dynamics of $f$ and $f_{\ast }$. For such a study, it is useful to consider the space ${\mathcal{C}}(K,X)$ of continuous maps from a Cantor set $K$ to $X$ provided with the topology of uniform convergence, and $f_{\ast }$ induced on ${\mathcal{C}}(K,X)$ by composition of maps. We mainly study the properties of transitive points of the induced system $(2^{X},f_{\ast })$ both topologically and dynamically, and give some examples. We also look into some more properties of the system $(2^{X},f_{\ast })$.

We construct the first examples of rational functions defined over a non-archimedean field with a certain dynamical property: the Julia set in the Berkovich projective line is connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we give an example for which the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.

Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semiconjugate to non-uniformly expanding maps. Using the theory of projective metrics on cones, we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from a solenoid, we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov.

We correct two technical errors in the original paper. The main result in the original paper remains valid without any changes.

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^{\ast }$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ from the graph algebra $C^{\ast }(E)$ and its diagonal subalgebra ${\mathcal{D}}(E)$ which generalises Renault’s Weyl groupoid construction applied to $(C^{\ast }(E),{\mathcal{D}}(E))$. We show that ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ recovers the graph groupoid ${\mathcal{G}}_{E}$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault’s results to $(C^{\ast }(E),{\mathcal{D}}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.