We introduce a correlation number for two strictly positive locally Hölder continuous independent potentials with strong entropy gaps at infinity on a topologically mixing countable state Markov shift with big images and pre-images (BIP) property. We define in this way a correlation number for pairs of cusped Hitchin representations. Furthermore, we explore the connection between the correlation number and the Manhattan curve, along with several rigidity properties of this correlation number.

We characterize dynamics of every distortion element in the group of diffeomorphisms of the 2-sphere that has at least two fixed points and another recurrent point. The key result is that if f is such a diffeomorphism, then the homeomorphism $\check {f}_{\mathrm { ann}}$, which is a lift of the homeomorphism of the closed annulus $\overline {\mathcal {A}}$ obtained from $\mathbb {S}^2$ by blowing up two fixed points of f to the universal covering space of $\overline {\mathcal {A}}$, has a unique rotation number. Moreover, we find the differential of such a distortion element in the group of diffeomorphisms of the 2-sphere at each fixed point up to conjugacy.

Orbit separation dimension ($\mathrm {OSD}$), previously introduced as amorphic complexity, is a powerful complexity measure for topological dynamical systems with pure-point spectrum. Here, we develop methods and tools for it that allow a systematic application to translation dynamical systems of tiling spaces that are generated by primitive inflation rules. These systems share many nice properties that permit the explicit computation of the $\mathrm {OSD}$, thus providing a rich class of examples with non-trivial $\mathrm {OSD}$.

We study the computational problem of rigorously describing the asymptotic behavior of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and we prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constraints and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non-computable.

We prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$, can be non-trivially and isometrically embedded in a family of piecewise isometries. In particular, this proves the existence of invariant curves for piecewise isometries, reminiscent of Kolmogorov–Arnold–Moser (KAM) curves for area-preserving maps, which are not unions of circle arcs or line segments.

Due to a result by Glasner and Downarowicz [Isomorphic extensions and applications. Topol. Methods Nonlinear Anal. 48(1) (2016), 321–338], it is known that a minimal system is mean equicontinuous if and only if it is an isomorphic extension of its maximal equicontinuous factor. The majority of known examples of this type are almost automorphic, that is, the factor map to the maximal equicontinuous factor is almost one-to-one. The only cases of isomorphic extensions which are not almost automorphic are again due to Glasner and Downarowicz, who in the same article provide a construction of such systems in a rather general topological setting. Here, we use the Anosov–Katok method to provide an alternative route to such examples and to show that these may be realized as smooth skew product diffeomorphisms of the two-torus with an irrational rotation on the base. Moreover – and more importantly – a modification of the construction allows to ensure that lifts of these diffeomorphisms to finite covering spaces provide novel examples of finite-to-one topomorphic extensions of irrational rotations. These are still strictly ergodic and share the same dynamical eigenvalues as the original system, but show an additional singular continuous component of the dynamical spectrum.

We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,\ldots , f_k$ such that the functions $f_i\circ T^n$, $n\in {\mathbb {Z}}$, $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.

In this paper, we reduce the logarithmic Sarnak conjecture to the $\{0,1\}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.

For a pseudo-Anosov flow $\varphi $ without perfect fits on a closed $3$-manifold, Agol–Guéritaud produce a veering triangulation $\tau $ on the manifold M obtained by deleting the singular orbits of $\varphi $. We show that $\tau $ can be realized in M so that its 2-skeleton is positively transverse to $\varphi $, and that the combinatorially defined flow graph $\Phi $ embedded in M uniformly codes the orbits of $\varphi $ in a precise sense. Together with these facts, we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of the closed orbits of $\varphi $ after cutting M along certain transverse surfaces, thereby generalizing the work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M. Our work can be used to study the flow $\varphi $ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the ‘positive’ cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.

Sarnak’s Möbius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $({1}/{N})\! \sum _{n=1}^{N}\! f(T^{n} x) \mu (n)= o(1)$ for every $f\in \mathcal {C}(X)$ and every $x\in X$. We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of $\mu (n)$, one can put any bounded sequence $a_{n}$ such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero. Moreover, in our construction, the choice of x depends on the sequence $a_{n}$ but $(X,T)$ does not.

We generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg–Zimmer tower of a measurably distal system and showing that at each step there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.

Given a dynamical simplex K on a Cantor space X, we consider the set $G_K^*$ of all homeomorphisms of X which preserve all elements of K and have no non-trivial clopen invariant subset. Generalizing a theorem of Yingst, we prove that for a generic element g of $G_K^*$ the set of invariant measures of g is equal to K. We also investigate when there exists a generic conjugacy class in $G_K^*$ and prove that this happens exactly when K has only one element, which is the unique invariant measure associated to some odometer; and that in that case the conjugacy class of this odometer is generic in $G_K^*$.

This paper presents sufficient conditions for a substitution tiling dynamical system of $\mathbb {R}^2$, generated by a generalized substitution on three letters, to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this paper are based on the work by Kenyon, Sadun, and Solomyak [Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys. 25(6) (2005), 1919–1934]. They studied one-dimensional tiling dynamical systems generated by substitutions on two letters and provided similar conditions sufficient to ensure that one-dimensional substitution tiling dynamical systems are topologically mixing. If a tiling dynamical system of $\mathbb {R}^2$ satisfies our conditions (and thus is topologically mixing), we can construct additional topologically mixing tiling dynamical systems of $\mathbb {R}^2$. By considering the stepped surface constructed from a tiling $T_\sigma $, we can get a new tiling of $\mathbb {R}^2$ by projecting the surface orthogonally onto an irrational plane through the origin.

We investigate to what extent a minimal topological dynamical system is uniquely determined by a set of return times to some open set. We show that in many situations, this is indeed the case as long as the closure of this open set has no non-trivial translational symmetries. For instance, we show that under this assumption, two Kronecker systems with the same set of return times must be isomorphic. More generally, we show that if a minimal dynamical system has a set of return times that coincides with a set of return times to some open set in a Kronecker system with translationarily asymmetric closure, then that Kronecker system must be a factor. We also study similar problems involving nilsystems and polynomial return times. We state a number of questions on whether these results extend to other homogeneous spaces and transitive group actions, some of which are already interesting for finite groups.

Let M be a geometrically finite acylindrical hyperbolic $3$-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J., to appear, Preprint, 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $3$-manifold $M_0$, the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$. We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.

Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$. In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

We prove that for $C^0$-generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. As an application, we show that the upper box dimension of the set of periodic points of a $C^0$-generic homeomorphism is equal to the dimension of the manifold. In the case of continuous interval maps, we prove that each level set for the metric mean dimension with respect to the Euclidean distance is $C^0$-dense in the space of continuous endomorphisms of $[0,1]$ with the uniform topology. Moreover, the maximum value is attained at a $C^0$-generic subset of continuous interval maps and a dense subset of metrics topologically equivalent to the Euclidean distance.

It is shown that the Ellis semigroup of a $\mathbb Z$-action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$- or $\mathbb R$-action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

We provide a complete characterization of periodic point free homeomorphisms of the $2$-torus admitting irrational circle rotations as topological factors. Given a homeomorphism of the $2$-torus without periodic points and exhibiting uniformly bounded rotational deviations with respect to a rational direction, we show that annularity and the geometry of its non-wandering set are the only possible obstructions for the existence of an irrational circle rotation as topological factor. Through a very precise study of the dynamics of the induced $\rho $-centralized skew-product, we extend and generalize considerably previous results of Jäger.