This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometries where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. Digital halftoning is a family of printing technologies for getting full-color images from only a few different colors deposited at dots all of the same size. The simplest version consists in obtaining gray-scale images from only black and white dots. A corollary of the theorem is that for error diffusion, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.

In a previous paper, some conditions under which the growth type is realizable as a growth type of a suitable Riemannian manifold were given. We show that a growth type satisfying these conditions can be realized as the growth type of an orbit of some finitely generated group of diffeomorphisms and as a growth type of a leaf of compact foliated manifold.

Let $\mathsf{X}_1$, $\mathsf{X}_2$ be mixing connected algebraic dynamical systems with the descending chain condition. We show that every equivariant continuous map $\mathsf{X}_1\to\mathsf{X}_2$ is affine (that is, $\mathsf{X}_2$ is topologically rigid) if and only if the system $\mathsf{X}_2$ has finite topological entropy.

We investigate the relation between preimage multiplicity and topological entropy for continuous maps. An argument originated by Misiurewicz and Przytycki shows that if every regular value of a C1 map has at least m preimages then the topological entropy of the map is at least log m. For every integer, there exist continuous maps of the circle with entropy 0 for which every point has at least m preimages. We show that if in addition there is a positive uniform lower bound on the diameter of all pointwise preimage sets, then the entropy is at least log m.

A graph surface P is a two-dimensional polyhedron having the simplest kind of non-trivial singularities which result from gluing surfaces with compact boundaries along boundary components. We study the behavior of the volume entropy h(g) of hyperbolic metrics g on a closed graph surface P depending on the lengths of singular geodesics $Q\subset P$. We show that always h(g) > 1 and $h(g)\to\infty$ as $L_g(Q)\to\infty$ for at least one singular geodesic Q.

For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost-sure central limit theorem (ASCLT). In fact, we provide a rate of convergence in the Kantorovich distance. Maxima of partial sums are also shown to obey an ASCLT. The key tool is an exponential inequality recently obtained. Then we establish (optimal) almost-sure convergence rates for the supremum of moving averages of Lipschitz observables (Erdös–Rényi-type law). This is done by refining the usual large-deviations estimates available for expanding maps of the interval. We end up with an application to entropy estimation ASCLTs that refine the Shannon–McMillan–Breiman and Ornstein–Weiss theorems.

We show that the Morse system is approximately transitive (AT) and that a system of positive entropy cannot be AT. We give examples of zero-entropy systems that are not AT, one of which is a two-point extension of a system that is AT.

We study the entropy of actions on function spaces with the focus on doubly stochastic operators on probability spaces and Markov operators on compact spaces. Using an axiomatic approach to entropy we prove that there is basically only one reasonable measure-theoretic entropy notion on doubly stochastic operators. By ‘reasonable’ we mean extending the Kolmogorov–Sinai entropy on measure-preserving transformations and satisfying some obvious continuity conditions for $H_\mu$. In particular, this establishes equality on such operators between the entropy notion introduced by R. Alicki, J. Andries, M. Fannes and P. Tuyls (a version of which was also studied by I. I. Makarov), another notion of entropy introduced by E. Ghys, R. Langevin and P. Walczak, and our new definition introduced later in this paper. The key tool in proving this uniqueness is the discovery of a very general property of all doubly stochastic operators, which we call asymptotic lattice stability. Unlike the other explicit definitions of entropy mentioned above, ours satisfies many natural requirements already on the level of the function $H_\mu$, and we prove that the limit defining $h_\mu$ exists. The proof uses an integral representation of a stochastic operator obtained many years ago by A. Iwanik. In the topological part of the paper we introduce three natural definitions of topological entropy for Markov operators on C(X). Then we prove that all three are equal. Finally, we establish the partial variational principle: the topological entropy of a Markov operator majorizes the measure-theoretic entropy of this operator with respect to any of its invariant probability measures.

We show that among three-interval exchange transformations there exists a dichotomy: T has minimal self-joinings whenever the associated subshift is linearly recurrent, and is rigid otherwise. We also build a family of simple rigid three-interval exchange transformations, which is a step towards an old question of Veech, and a family of rigid three-interval exchange transformations, which includes Katok's rank-one map.

We study here the action of subgroups of PSL$(2,\mathbb{R})$ on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem. Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on the manifold, is the function leafwise constant? We give a number of positive results and also show a general class of examples for which the Liouville property does not hold. The connection between the Liouville property and the dynamics on the space of harmonic functions as well as general properties of this dynamical system are explored. It is shown among other properties that the $\mathbb{Z}$-action generated by hyperbolic or parabolic elements of PSL$(2,\mathbb{R})$ is chaotic.

We obtain sharp results for the genericity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the Cr-topology for all r > 0 (here r need not be an integer and C1 is replaced by Lipschitz). Moreover, when $r\ge2$, we show that there is a C2-open and Cr-dense subset of Cr-extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) $\mathbb{R}^m$-extensions, thereby generalizing a result of Niţică and Pollicott, and on stable mixing for suspension flows.

A one-parameter family of circle homeomorphisms with several break-type singularities is considered. If one break dominates, in some sense, all the others, the rotation number is rational for almost all parameter values. Results for two break-type singularities are applied to a geometric problem and yield the typicality of periodic trajectories.

In this paper we show that two semi-conjugate random dynamical systems (RDSs) on Polish spaces have the same entropy and pressure if the cardinal number of the pre-image of a point under the semi-conjugacy is finite almost everywhere.

Let $(M,\{\phi^t\})$ be a smooth (not necessarily transitive) Anosov flow without fixed points generated by a vector field $X(x)=(d/dt)|_{t=0}\phi^t(x)$ on a compact manifold M. Let $A:M\rightarrow\mathbb{R}$ be a globally Hölder function defined on M. Assume that $\int_0^T A\circ\phi^t(x)\,dt\geq0$ for any periodic orbit $\{\phi^t(x)\}_{t=0}^{t=T}$ of period T. Then there exists a Hölder function $V:M\rightarrow \mathbb{R}$, called a sub-action, smooth in the flow direction, such that

\[A(x)\geq L_XV(x),\quad\text{for all }x\in M\]

(where $L_XV=(d/dt)|_{t=0}V\circ\phi^t(x)$ denotes the Lie derivative of V). If A is $\mathcal{C}^r$ then LXV is $\mathcal{C}^r$ on any local center-stable manifold.

In this paper we show that a continuous map f from a connected graph G to itself is pointwise-recurrent if and only if one of the following two statements holds: (1) G is a circle, and f is a homeomorphism topologically conjugate to an irrational rotation of the unit circle S1; (2) f is a periodic homeomorphism.