In this paper, we establish analytic conjugacy theorems of Poincaré type and Poincaré–Dulac type for analytic random dynamical systems based on their Lyapunov exponents.

We study the multifractal analysis of the pointwise dimension for equilibrium measures on countable Markov shifts. The main difficulty is that the space is not compact. In order to overcome this, we use an approximation argument based on the theory of convergence of Fenchel pairs developed by Wijsman. The results of Pesin and Weiss on multifractal analysis for compact spaces are used as well. We also prove a Bowen formula for countable Markov shifts. It turns out that, in this setting, this formula provides the Hausdorff dimension of the set of recurrent points.

Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of $\mathbb{Z}^{d}$ by translations on X is uniformly distributed on the union of several sub-nilmanifolds of X. As a corollary we obtain the pointwise ergodic theorem for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold.

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.

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.

We build a Markov partition of the Feigenbaum dynamical plane, and a system of ‘external rays’, which are invariant under rescaling. Applications to geometry of the Feigenbaum map are given.

In this paper we extend the results of an earlier paper, which deal with a description of the smooth untwisted cohomology for ${\mathbb Z}^k$-actions by hyperbolic automorphisms of a torus, to the partially hyperbolic case. Along the way we correct an error found in one of the steps in the proof for the hyperbolic case.

Hill's lunar problem appears in celestial mechanics as a limit of the restricted three-body problem. It is parameter-free and thus globally far from any simple well-known problem, and has shed strong numerical evidence of its lack of integrability in the past. An algebraic proof of meromorphic non-integrability is presented here. Beyond the result itself, the paper can also be considered as an example of the application of differential Galois and Morales–Ramis theories to a significant problem.

In this work, singular orientable foliations which admit non-trivial recurrent leaves on 2-manifolds of finite or infinite genus are considered. A structure theorem for such foliations is given. The theorem is similar to Gutierrez's structure theorem (C. Gutierrez. Smoothing continuous flows on 2-manifolds and recurrences. Ergod. Th. & Dynam. Sys.6 (1986), 17–44) for flows on compact surfaces.

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.

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.

We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a two-symbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing.

We prove pointwise ergodic theorems for general families of radial averages on semi-simple Lie groups, as well as strong Lp-maximal inequalities for the corresponding maximal functions, for p > 1. The methods of proof make essential use of exponential volume growth on the group, and of spectral considerations utilizing the existence of spectral gaps in measure-preserving actions of simple groups of real rank at least two. We also consider the case of the action by convolutions on the symmetric space, where we establish maximal inequalities of weak-type, going beyond Lp, p > 1. The proof here is based on some new convolution inequalities which are established by direct geometric comparison arguments. These inequalities can be viewed as a quantitative version of the wave front lemma. The maximal inequalities and pointwise ergodic theorems we establish are in sharp contrast to what one would be lead to expect by comparison with the analogous averages on the Abelian groups $\mathbb{R}^n$. We explain how the contrast arises from the difference between the polynomial volume growth on $\mathbb{R}^n$ and the exponential volume growth on semi-simple groups.

We consider the class of $\mathbb{R}^n$ extensions of Anosov diffeomorphisms on infranilmanifolds, and find necessary and sufficient conditions for topological transitivity. In particular, if the fiber is $\mathbb{R}$, the existence of a semi-orbit with the projection on $\mathbb{R}$ unbounded from above and from below is equivalent to topological transitivity. We also show that in the above class topological transitivity and stable topological transitivity are equivalent.

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.

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.

We will give a sufficient condition for points in a locally defined analytic subvariety V in $\mathbb{P}^k$ to be contained in the Fatou set in V for a birational self-map of degree ${\ge} 2$ of $\mathbb{P}^k$. We will consider its applications.

We investigate topological mixing for $\mathbb Z$ and $\mathbb R$ actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $\theta_2$ of the substitution matrix satisfies $|\theta_2|\ne 1$. If $|\theta_2|<1$, then (as is well known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|\theta_2|> 1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|\theta_2|=1$ is more delicate, and we only obtain some partial results.

Let $f: M \to M$ be a diffeomorphism defined in a three-dimensional compact boundary-less manifold M. We prove that for an open dense set, C1-robustly expansive homoclinic classes H(p) for f are hyperbolic. A diffeomorphism f is $\alpha$-expansive on a compact invariant set K if there is $\alpha>0$ such that for all $x,y\in K$, if ${\rm dist}(f^n(x),f^n(y))\leq \alpha$ for all $n\in \mathbb Z$ then x = y. By ‘robustly’ we mean that there is $\alpha>0$ such that for all nearby diffeomorphisms g, the homoclinic class H(pg) of the continuation of p is $\alpha$-expansive.

Consider the three-body problem with an attractive 1/r2 potential. Modulo symmetries, the dynamics of the bounded zero-angular-momentum solutions is equivalent to a geodesic flow on the thrice-punctured sphere, or ‘pair of pants’. The sphere is the shape sphere. The punctures are the binary collisions. The metric generating the geodesics is the Jacobi–Maupertuis metric. It is complete, has infinite area, and its ends, the neighborhoods of the punctures, are asymptotically cylindrical. Our main result is that when the three masses are equal then the metric has negative curvature everywhere except at two points (the Lagrange points). A corollary of this negativity is the uniqueness of the 1/r2 figure-eight, a complete symbolic dynamics for encoding the collision-free solutions, and the fact that collision solutions are dense within the bound solutions.