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.

We characterize the set of periods for tree maps. More precisely, we prove that the set of periods of any tree map $f:T \to T$ is the union of finitely many initial segments of Baldwin's orderings $_p{\geq}$ and a finite set $\mathcal{F}$. The possible values of p and explicit upper bounds for the size of $\mathcal{F}$ are given in terms of the combinatorial properties of the tree T. Conversely, given any set $\mathcal{A}$ which is a union of finitely many initial segments of Baldwin's orderings $_p{\geq}$ with p of the above type and a finite set, we prove that there exists a tree map whose set of periods is $\mathcal{A}$.

We continue studying branched covering maps of the plane with expanding properties, which we call expanding polymodials. They are analogous to piecewise expanding interval maps and have properties similar to those of complex polynomials (in particular, the Julia set and a lot of other notions from complex dynamics can be defined for expanding polymodials). In this paper we include the case when the Julia set is disconnected, study its topological properties and its Hausdorff dimension.

Lyapunov graphs carry dynamical information of gradient-like flows as well as topological information of their phase space which is taken to be a closed orientable n-manifold. In this paper we will show that an abstract Lyapunov graph $L(h_0, \dotsc,h_n,\kappa)$ in dimension n greater than 2, with cycle number $\kappa$, satisfies the Poincaré–Hopf inequalities if and only if it satisfies the Morse inequalities and the first Betti number, $\gamma_1$, is greater than or equal to $\kappa$. We also show a continuation theorem for abstract Lyapunov graphs with the presence of cycles. Finally, a family of Lyapunov graphs $\mathcal{L}(h_0, \dotsc, h_n,\kappa)$ with fixed pre-assigned data $(h_0, \dotsc, h_n,\kappa)$ is associated with the Morse polytope, $\mathcal{P}_{\kappa}(h_0,\dotsc, h_n)$, determined by the Morse inequalities for the given data.

In this paper we consider twist mappings of the torus, $\overline{T}:{\rm T^2\rightarrow T^2}$, and their vertical rotation intervals $\rho _V(T)=[\rho _V^{-},\rho _V^{+}]$, which are closed intervals such that for any $\omega \in\, ]\rho _V^{-},\rho _V^{+}[$ there exists a compact $\overline{T}$-invariant set $\overline{Q}_\omega $ with $\rho _V(\overline{x})=\omega$ for any $\overline{x}\in \overline{Q}_\omega $, where $\rho _V(\overline{x})$ is the vertical rotation number of $\overline{x}$. In the case when $\omega $ is a rational number, $\overline{Q}_\omega $ is a periodic orbit. Here we analyze how $\rho _V^{-}$ and $\rho _V^{+}$ behave as we perturb $\overline{T}$ and which dynamical properties for $\overline{T}$ can be obtained from their values.

Assume that T is a conservative ergodic measure-preserving transformation of the infinite measure space $(X,\mathcal{A},\mu)$. We study the asymptotic behaviour of occupation times of certain subsets of infinite measure. Specifically, we prove a Darling–Kac type distributional limit theorem for occupation times of barely infinite components which are separated from the rest of the space by a set of finite measure with continued-fraction (CF)-mixing return process. In the same setup we show that the ratios of occupation times of two components separated in this way diverge almost everywhere. These abstract results are illustrated by applications to interval maps with indifferent fixed points.

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.

We extend the Delorme–Guichardet characterization of Kazhdan property T groups to r-discrete measured groupoids. We give several applications, in particular to the stability results of Kazhdan property T groupoids and to the study of cocycles taking values in a group having the Haagerup property.

Nous montrons que C1-génériquement les difféomorphismes symplectiques d'une variété compacte sont transitifs. Sur les variétés symplectiques non-compactes, nous montrons que les orbites génériques d'un difféomorphisme générique ne sont pas bornées. L'outil principal est un lemme de connexion pour les pseudo-orbites des difféomorphismes symplectiques. Pour cela nous donnons des conditions générales sur un espace de difféomorphismes, en particulier au voisinage des orbites périodiques, qui permettent d'obtenir le lemme de connexion pour les pseudo-orbites.

We show that the renewal theory developed by Sarig and Gouëzel in the context of non-uniformly expanding dynamical systems applies also to the study of compact group extensions of such systems. As a consequence, we obtain results on subexponential decay of correlations for equivariant Hölder observations.

We prove that a minimal self-mapping of a compact 2-manifold has tree-like fibers (i.e. all points have preimages which are connected, at most one-dimensional and with trivial shape). We also prove that the only 2-manifolds (compact or not) which admit minimal maps are either finite unions of tori, or finite unions of Klein bottles.

We introduce the notion of an Anosov family, a generalization of an Anosov map of a manifold. This is a sequence of diffeomorphisms along compact Riemannian manifolds such that the tangent bundles split into expanding and contracting subspaces. We develop the general theory, studying sequences of maps up to a notion of isomorphism and with respect to an equivalence relation generated by two natural operations, gathering and dispersal. Then we concentrate on linear Anosov families on the 2-torus. We study in detail a basic class of examples, the multiplicative families, and a canonical dispersal of these, the additive families. These form a natural completion to the collection of all linear Anosov maps. A renormalization procedure constructs a sequence of Markov partitions consisting of two rectangles for a given additive family. This codes the family by the non-stationary subshift of finite type determined by exactly the same sequence of matrices. Any linear positive Anosov family on the torus has a dispersal which is an additive family. The additive coding then yields a combinatorial model for the linear family, by telescoping the additive Bratteli diagram. The resulting combinatorial space is then determined by the same sequence of non-negative matrices, as a non-stationary edge shift. This generalizes and provides a new proof for theorems of Adler and Manning.

We prove measurable Livšic theorems for dynamical systems modelled by Markov towers. Our regularity results apply to solutions of cohomological equations posed on Hénon-like mappings and a wide variety of non-uniformly hyperbolic systems. We consider both Hölder cocycles and cocycles with singularities of prescribed order.

Transfer operators ${\mathcal M}_k$ are associated to Cr transversal local diffeomorphisms $\psi_\omega$ of ${\mathbb R}^n$, and Cr compactly supported functions $g_\omega$. A formal trace $\operatorname{tr}^\# {\mathcal M}$, yields a formal Ruelle–Lefschetz determinant $\operatorname{Det}^\#(\operatorname{Id} -z{\mathcal M})$. We use the Milnor–Thurston–Kitaev equality recently proved by Baillif to relate zeros and poles of $\operatorname{Det}^\#(\operatorname{Id} -z{\mathcal M})$ with spectra of the transfer operators ${\mathcal M}_k$, under additional assumptions. As an application, we obtain a new proof of a result of Ruelle on the spectral interpretation of zeros and poles of the dynamical zeta function $\exp \sum_{m\ge1}(z^m/m) \sum_{f^m (x)=x} |{\rm det}\, Df(x)|^{-1}$ for smooth expanding endomorphisms f.

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 study a d-dimensional coupled map lattice consisting of hyperbolic toral automorphisms (Arnold cat maps) that are weakly coupled by an analytic map. We construct the Sinai–Ruelle–Bowen measure for this system and study its marginals on the tori. We prove that they are absolutely continuous with respect to Lebesgue measure if and only if the coupling satisfies a non-degeneracy condition.

We study the discrete quasi-periodic Schrödinger equation \[-(u_{n+1}+u_{n-1})+\lambda V(\theta+n\omega)u_n=Eu_n\] with a non-constant C1 potential function $V:\mathbb{T}\to\mathbb{R}$. We prove that for sufficiently large $\lambda$ there is a set $\Omega\subset\mathbb{T}$ of frequencies $\omega$, whose measure tends to 1 as $\lambda\to\infty$, with the following property. For each $\omega\in\Omega$ there is a ‘large’ (in measure) set of energies E, all lying in the spectrum of the associated Schrödinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrödinger cocycle is minimal but not ergodic.

Let X be a graph and let G be a subgroup of the automorphism group of X. We investigate the question of when the full 2-shift process on the vertices of X has a G-factor process on the same graph consisting of random variables which are independent and identically distributed (i.i.d.) uniform in [0, 1]. We show that such a factor always exists when X is a tree with bounded degrees, no leaves and at least three topological ends, and give a sufficient condition for when such a factor exists if X is a connected graph with infinitely many ends and G acts transitively on the vertices of X. We also show that the full 2m-shift on any finitely-generated non-abelian group G has a G-factor which is i.i.d. uniform on [0, 1] when m is sufficiently large.

We determine the Hausdorff and box dimensions of the flexed Sierpiński gasket $\Lambda$, which is the limit set for a non-Moran-like non-affine geometric construction in the plane. We also provide a multifractal formalism for the Bernoulli measures on $\Lambda$, computing explicitly the pointwise dimension spectra.

In this paper we obtain (almost) optimal results concerning randomly weighted one-sided ergodic Hilbert transforms. Given an iid sequence of centered random variables (Xn) in L log L, we show that there exists a universal set $\Omega'$ of probability 1 such that for any $\omega\in \Omega'$ the realization $(X_n(\omega))$ is good for the almost everywhere convergence of the weighted one-sided ergodic Hilbert transform associated with any dynamical system and any $g\in L\log L$. The method applies to powers along subsequences with ‘small’ growth and when considering Dunford–Schwartz operators instead of pointwise transformation. If the (Xn) are symmetric, but only in L log log L, we obtain a slightly weaker result.