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 $\mathcal{A}$.

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.

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.

A result of Krengel on the density of the set of vectors with infinitely many pairwise orthogonal iterates under powers of a unitary operator with continuous spectrum is generalized to actions of countable Abelian groups.

A block code from an irreducible shift of finite type can be lifted canonically through resolving maps to a resolving map. There is an application to Markovian maps.

We realize the Hecke C*-algebra $\mathcal{C}_{\mathbb{Q}}$ of Bost and Connes as a direct limit of Hecke C*-algebras which are semigroup crossed products by $\mathbb{N}^F$, for F a finite set of primes. For each approximating Hecke C*-algebra we describe a composition series of ideals. In all cases there is a large type I ideal and a commutative quotient, and the intermediate subquotients are direct sums of simple C*-algebras. We can describe the simple summands as ordinary crossed products by actions of $\mathbb{Z}^S$ for S a finite set of primes. When $\vert S\vert =1$, these actions are odometers and the crossed products are Bunce–Deddens algebras; when $\vert S\vert >1$, the actions are an apparently new class of higher-rank odometer actions, and the crossed products are an apparently new class of classifiable AT-algebras.

For locally compact groups all actions on a standard measure algebra have a spatial realization. For many Polish groups this is no longer the case. However, we show here that for non-archimedean Polish groups all measure algebra actions do have spatial realizations. In the other direction we show that an action of a Polish group is whirly (‘ergodic at the identity’) if and only if it admits no spatial factors and that all actions of a Lévy group are whirly. We also show that in the Polish group $\rm{Aut\,}(X,\mathcal{X},\mu)$, for the generic automorphism T the action of the subgroup $\Lambda(T)=\rm{cls\,}\{T^n: n\in \mathbb{Z}\}$ on the Lebesgue space $(X,\mathcal{X},\mu)$ is whirly.

The system $-d^2y/dt^2+u(t)y=Ey\ (y\in R^n)$ is considered, where $E=\text{diag} (\lambda_1^2,\, \lambda_2^2,\dots,\lambda_n^2)$ is a diagonal matrix with $\lambda_j>0\ (j=1,2,\dots, n)$ being regarded as parameters, and u a real analytic quasi-periodic symmetric matrix. Suppose that the basic frequencies of u satisfy a Diophantine condition. Then, for most of sufficiently large $\lambda_j\ (j=1,2,\dots,n)$, the system has n pairs of conjugate quasi-periodic solutions.

We will deal with two topics about normality of the iterates of holomorphic self-maps in compact varieties. First, for a holomorphic self-map f of an irreducible compact variety Z, we show that if at least one subsequence of $\{f^n\}_{n\ge0}$ converges uniformly, then the full sequence $\{f^n\}_{n\ge0}$ itself is a normal family and the full set of the limit maps is finite or has the structure of a compact commutative Lie group. Second, in the case when Z is non-singular, we deal with the dynamics on forward-invariant compact subsets outside the closures of the post-critical sets. We will describe the semi-repelling structure of the dynamics in terms of repelling points and neutral Fatou discs (center manifolds). In particular, in the case of holomorphic self-maps of projective spaces, we will obtain a stronger result.

Let R be a rational map. A critical point c is called summable if the series $\sum_i(1/(R^i)'(R(c)))$ is absolutely convergent. Under certain topological conditions on the postcritical set we prove that R cannot be structurally stable if it has a summable critical point $c \in J(R)$.

If $\mathbb{L} =\mathbb{Z}^D$ and $\mathcal{A}$ is a finite set, then $\mathcal{A}^\mathbb{L}$ is a compact space; a cellular automaton (CA) is a continuous transformation $\Phi:\mathcal{A}^\mathbb{L} \longrightarrow\mathcal{A}^\mathbb{L}$ that commutes with all shift maps. A quasisturmian (QS) subshift is a shift-invariant subset obtained by mapping the trajectories of an irrational torus rotation through a partition of the torus. The image of a QS shift under a CA is again QS. We study the topological dynamical properties of CA restricted to QS shifts, and compare them to the properties of CA on the full shift $\mathcal{A}^\mathbb{L}$. We investigate injectivity, surjectivity, transitivity, expansiveness, rigidity, fixed/periodic points and invariant measures. We also study ‘chopping’: how iterating the CA fragments the partition generating the QS shift.

In this paper, we address the positive recurrent diffusion process obtained by winding the solutions of a general stochastic system of coupled oscillators along several directions. Two kinds of quantities—the entropy production rate and the rotation numbers, which describe the non-equilibrium of the system from totally different points of view—are calculated and related through an explicit formula.

We study the Radon–Nikodym problem for approximately proper equivalence relations and more specifically the uniqueness of certain Gibbs states. One of our tools is a variant of the dimension group introduced in the study of approximately finite-dimensional (AF) algebras. As applications, we retrieve sufficient conditions for the uniqueness of traces on AF algebras and parts of the Perron–Frobenius–Ruelle theorem.

An example of a two-dimensional subshift is the set of arrays $\{x_{(i,j)}:(i,j)\in\mathbb{Z}^2\}$ of zeros and ones for which $x_{(i,j)}+x_{(i+1,j)}+x_{(i,j+1)}=0$ for each $i,j\in\mathbb{Z}$, where addition is taken modulo 2. There are two shift operators, the horizontal shift and the vertical shift. This subshift is called the Ledrappier example or the three dot dynamical system. In this paper we give an internal description of all transitive subshifts for which one of the shift operators is sofic in the usual one-dimensional sense.