First, the class of id-finitely fixed actions of ℤd on a Lebesgue space is defined. Then, it is demonstrated that this property is stable under id-Kakutani equivalence and that, conversely, any two id-finitely fixed ℤd-actions of the same (finite) positive entropy are id-Kakutani equivalent. By id-Kakutani equivalence we mean that element in A. del Junco and D. Rudolph's family of relations on ℤd-actions corresponding to the identity d × d matrix.

We show that a C0 minimal action of a semisimple Lie group without compact factors is either locally free or induced from a minimal action of a proper parabolic subgroup. We describe the orbit structure of the action restricted to a maximal compact subgroup, and then apply this to minimal actions in low dimension. We give some examples, applications, and open problems.

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

Let X be a compact, connected subset of a smooth manifold. Suppose that X admits a codimension d lamination, d\ge 2. Let x,y\in X and \epsilon>0. There exists a sequence x=x_0,\dotsc,x_k=y in X such that the sum of the dth powers of the distances between successive points is less than \epsilon. We discuss two proposed applications of this result.

Kifer and Ledrappier have asked whether the harmonic measures {νx} on manifolds of negative curvature are equivalent to the conditional measures of the harmonic measure v of the geodesic flow associated with the fibration {SxM}x∈M. We settle this question with a rigidity result. We also clear up the same problem concerning the Patterson-Sullivan measure and the Bowen–Margulis measure.

In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

In this paper, we construct and study a semigroup associated to an action of a countable discrete group on a compact Hausdorff space that can be regarded as a higher dimensional generalization of the type semigroup. We study when this semigroup is almost unperforated. This leads to a new characterization of dynamical comparison and thus answers a question of Kerr and Schafhauser. In addition, this paper suggests a definition of comparison for dynamical systems in which neither the acting group is necessarily amenable nor the action is minimal.

We show relations between new notions on cellular automata based on probabilistic dynamics: almost everywhere sensitivity to initial conditions for Besicovitch pseudo-distance, damage spreading (which measures the information (or damage) propagation), and the destruction of the initial configuration information. Through natural examples, we illustrate the relations between these formal definitions.

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

In this paper, we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection. We show that in the case of binary words, the bound is actually achieved and gives a characterization of recurrent aperiodic words.

We obtain a unique, canonical one-to-one correspondence between the space of marked postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of entire maps that take the form $p(z)\exp (q(z))$ for $p(z)$, $q(z)$ polynomials and $\exp (z)$, the complex exponential function. This bijection preserves the dynamics and embedding of Julia sets and is induced by a surgery tool developed by Haïssinsky.

In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on ${\mathbb T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.

We consider measurable conjugacies between two non-solvable subgroups G_1, G_2 of the group of analytic diffeomorphisms of the circle which admit a finite generating set close to the identity. These measurable conjugacies are characterized by our main result which asserts that, up to finite coverings, they coincide almost everywhere with an analytic diffeomorphism of the circle.

In the present paper we introduce positive flows and processes, which generalize ordinary dynamical systems and stochastic processes. The theory of positive flows and processes is based on the methods of positive operator theory and operator algebras theory. Its basic concepts are the phase and positive algebras, the spectral potential, dual entropy, equilibrium measures, action functional, sensitive states, and empirical measures. Within the frames of the theory we prove the law of large numbers with respect to the sensitive states and calculate asymptotics for the probabilities of large deviations in terms of the action functional.

We show that every locally finite random graph embedded in the plane with an isometry-invariant distribution can be five-colored in an invariant and deterministic way, under some non-triviality assumption and a mild assumption on the tail of edge lengths. The assumptions hold for any Voronoi map on a point process that has no non-trivial symmetries almost surely, hence we improve and generalize previous results on six-coloring the Voronoi map on a Poisson point process (see Angel, Benjamini, Gurel-Gurevich, Mayerovitch and Peled [Stationary map coloring. Preprint, 2008]).

We consider the deviation of Birkhoff sums along fixed orbits of substitution dynamical systems. We show distributional convergence for the Birkhoff sums of eigenfunctions of the substitution matrix. For non-coboundary eigenfunctions with eigenvalue of modulus $1$, we obtain a central limit theorem. For other eigenfunctions, we show convergence to distributions supported on Cantor sets. We also give a new criterion for such an eigenfunction to be a coboundary, as well as a new characterization of substitution dynamical systems with bounded discrepancy.

Let $(f,\alpha)$ be the process given by an endomorphism $f$ and by a finite partition $\alpha=\{A\}_{i=1}^{s}$ of a Lebesgue space. Let $E(f,\alpha)$ be the set of densities of absolutely continuous invariant measures for skew products with the base $(f,\alpha)$. We prove that a process $(f,\alpha)$ is Markovian if and only if $E(f,\alpha)\subset \{g: g=\sum_{i=1}^{s} 1_{A_{i}}\otimes g_{i}\}$. If $f$ is the Lasota–Yorke or Misiurewicz type map with the Markovian partition $\alpha$, then $(f,\alpha)$ is quasi-Markovian, i.e. $E(f,\alpha)\subset \{g:\supp g=\bigcup_{i=1}^{s} A_{i}\times B_{i}\}$. Moreover, we give the characterization of quasi-Markovian processes.

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.

We study the dynamics of the map $f:\mathbb {A}^N\to \mathbb {A}^N$ defined by

$$ \begin{align*} f(\mathbf{X})=A\mathbf{X}^d+\mathbf{b}, \end{align*} $$

for $A\in \operatorname {SL}_N$ , $\mathbf {b}\in \mathbb {A}^N$ , and $d\geq 2$ , a class which specializes to the unicritical polynomials when $N=1$ . In the case $k=\mathbb {C}$ we obtain lower bounds on the sum of Lyapunov exponents of f, and a statement which generalizes the compactness of the Mandelbrot set. Over $\overline {\mathbb {Q}}$ we obtain estimates on the critical height of f, and over algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.

For a transitive countably piecewise monotone Markov interval map we consider the question of whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).