In the 1970s, Nekhorochev proved that, for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Recently, Guzzo has given examples of exponentially stable integrable Hamiltonians that are non-steep but satisfy a weak condition of transversality which involves only the affine subspaces spanned by integer vectors. We generalize this notion for an arbitrary integrable Hamiltonian and prove Nekhorochev's estimates in this setting. The point in this refinement lies in the fact that it allows one to exhibit a generic class of real analytic integrable Hamiltonians which are exponentially stable with fixed exponents. Genericity is proved in the sense of measure since we exhibit a prevalent set of integrable Hamiltonians which satisfy the latter property. This is obtained by an application of a quantitative Sard theorem given by Yomdin.

This paper is concerned with Borel 1-cocycles on ergodic flows on standard Borel measure spaces and a certain type of group coactions on (separable) factors.

Let be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the -action of F and the shift map σ, to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group . This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12(4) (2005), 723–736). This result is applied to give conditions which also force an (F,σ)-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on .

Let $(X,d,T)$ be a dynamical system, where $(X,d)$ is a compact metric space and $T:X\rightarrow X$ a continuous map. We introduce two conditions for the set of orbits, called respectively the $\texttt{g}$-almost product property and the uniform separation property. The $\texttt{g}$-almost product property holds for dynamical systems with the specification property, but also for many others. For example, all $\beta$-shifts have the $\texttt{g}$-almost product property. The uniform separation property is true for expansive and more generally asymptotically $h$-expansive maps. Under these two conditions we compute the topological entropy of saturated sets. If the uniform separation condition does not hold, then we can compute the topological entropy of the set of generic points, and show that for any invariant probability measure $\mu$, the (metric) entropy of $\mu$ is equal to the topological entropy of generic points of $\mu$. We give an application of these results to multi-fractal analysis and compare our results with those of Takens and Verbitskiy (Ergod. Th. & Dynam. Sys.23 (2003), 317–348).

First notions of relative complexity function and relative sensitivity are introduced. It turns out that for any open factor map $\pi: (X, T)\rightarrow (Y, S)$ between topological dynamical systems with minimal $(Y, S),\ \pi$ is positively equicontinuous if and only if the relative complexity function is bounded for each open cover of $X$; and that any non-trivial weakly mixing extension is relatively sensitive. Moreover, a relative version of the notable result that any $M$-system is sensitive if it is not minimal is obtained. Then notions of relative scattering and relative Mycielski's chaos are introduced. A relative disjointness theorem involving relative scattering is given. A relative version of the well-known result that any non-trivial scattering topological dynamical system is Li–Yorke chaotic is proved.

We relate Kellendonk and Putnam’s pattern-equivariant (PE) cohomology to the inverse-limit structure of a tiling space. This gives a version of PE cohomology with integer coefficients, or with values in any Abelian group. It also provides an easy proof of Kellendonk and Putnam’s original theorem relating PE cohomology to the Čech cohomology of the tiling space. The inverse-limit structure also allows for the construction of a new non-Abelian invariant, the PE representation variety.

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.

In this paper we will show that, for any strong orbit equivalence class for Cantor minimal systems, the class contains (minimal) subshifts which attain arbitrary finite topological entropy.

We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.

We define and study a new invariant called pre-image pressure and its relationship with invariant measures. More precisely, for a given dynamical system $(X,f)$ (where $X$ is a compact metric space and $f$ is a continuous map from $X$ to itself) and $\varphi\in C(X,R)$ (the space of real-valued continuous functions on $X$), we prove a variational principle for pre-image pressure $P_{\rm pre}(f,\varphi),\ P_{\rm pre}(f,\varphi) = \sup_{\mu \in \mathcal{M}(f)} \{ h_{{\rm pre},\mu}(f) + \int \varphi \,d\mu\}$, where $h_{{\rm pre},\mu}(f)$ is the pre-image entropy (W.-C. Cheng and S. Newhouse. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113) and $\mathcal{M}(f)$ is the set of invariant measures of $f$. Moreover, we also prove that pre-image pressure determines the invariant measures and give some applications of pre-image pressure to equilibrium states.