A dynamical criterion of conjugacy of a flow with a shift and a counterpart of the Brouwer plane translation theorem for flows on a normal topological space are established.

In a topological dynamical system $(X,T)$ the complexity function of a cover ${\cal C}$ is the minimal cardinality of a sub-cover of $\bigvee_{i=0}^n T^{-i}{\cal C}$. It is shown that equicontinuous transformations are exactly those such that any open cover has bounded complexity. Call scattering a system such that any finite cover by non-dense open sets has unbounded complexity, and call 2-scattering a system such that any such 2-set cover has unbounded complexity: then all weakly mixing systems are scattering and all 2-scattering systems are totally transitive. Conversely, any system that is not 2-scattering has covers with complexity at most $n+1$. Scattering systems are characterized topologically as those such that their cartesian product with any minimal system is transitive; they are consequently disjoint from all minimal distal systems. Finally, defining $(x,y)$, $x\ne y$, to be a complexity pair if any cover by two non-trivial closed sets separating $x$ from $y$ has unbounded complexity, we prove that 2-scattering systems are disjoint from minimal isometries; that in the invertible case the complexity relation is contained in the regionally proximal relation and, when further assuming minimality, coincides with it up to the diagonal.

We prove that a relevant part of the Lyapunov spectrum and the corresponding Oseledets spaces of a quasi-compact linear cocycle are stable under a certain type of random perturbation. The basic approach is a graph-transform argument. The result applies to the spectrum of (not necessarily i.i.d.) randomly perturbed expanding maps and yields generalizations of results recently obtained by Baladi, Kondah and Schmitt [5] with different methods.

A one-parameter family of time reversible Anosov flows is studied; physically, it describes a particle moving on a surface of constant negative curvature under the action of an electric field (corresponding to an automorphic form) and of a ‘thermostatting’ force (given by Gauss's least-constraint principle). We show that the flows are dissipative, in the sense that the average volume contraction rate is positive and the Sinai–Ruelle–Bowen measure is singular with respect to the volume: therefore they verify the assumptions for the validity of the continuous time version of Gallavotti–Cohen's fluctuation theorem for the large fluctuations of the average volume contraction rate. If several independent electric fields are considered, it makes sense to ask for the validity of Onsager's reciprocity: we show, by explicitly computing the relevant transport coefficents, that it is indeed obeyed.

We prove that any expanding piecewise real-analytic map of a bounded region of the plane admits absolutely continuous invariant probability measures.

The joint study of functionals of a Brownian motion $B$ and its Lévy transform $\beta= |B|-L$, where $L$ is the local time of $B$ at zero, is motivated by the conjectured ergodicity of the Lévy transform.

Here, we compute explicitly the covariance of the last zeros before time one of $B$ and $\beta$, which turns out to be strictly positive.

Let $\mathcal{H}$ be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that $\mathcal{H}$ has compact closure in moduli space if and only if neither attractor is a fixed point.

We consider the dynamics of piecewise smooth interval maps $f$ with a nowhere vanishing derivative. We show that if $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of $f$ is expanding. In this case, $f$ admits an absolutely continuous invariant probability measure.

We characterize the Hamiltonian flows in the plane which are structurally stable (in a global sense) among Hamiltonian flows. This notion is closely related to, but distinct from, the topological stability of the generating function as a map from the plane to the line.

In this paper we prove that permutations of even length with maximal entropy are not unique. There are in fact two distinct permutations, $\theta_n$ and $\widetilde{\theta_n}$, with maximal entropy, both of which are self-dual and $\widetilde{\theta_n}$ is the reverse of $\theta_n$.

Let $G$ be a finitely generated nilpotent group of unitary operators on a Hilbert space ${\cal H}$. We prove that ${\cal H}$ is decomposable into a direct sum ${\cal H}=\bigoplus_{\alpha\in A}{\cal L}_{\alpha}$ of pairwise orthogonal closed subspaces so that elements of $G$ permute the subspaces ${\cal L}_{\alpha}$, and if $T({\cal L}_{\alpha})={\cal L}_{\alpha}$, then the action of $T$ on ${\cal L}_{\alpha}$ is either scalar or has continuous spectrum. We also provide examples showing that analogous results do not hold for solvable non-nilpotent groups.

We construct stabilized $C^*$-algebras from subshifts by using the dynamical property of the symbolic dynamical systems. We prove that the construction is dynamical and the $C^*$-algebras are isomorphic to the tensor product $C^*$-algebras between the algebra of all compact operators on a separable Hilbert space and the $C^*$-algebras constructed from creation operators on sub-Fock spaces associated with the subshifts. We also prove that the gauge actions on the stabilized $C^*$-algebras are invariant for topological conjugacy as two-sided subshifts under some conditions. Hence, if two subshifts are topologically conjugate as two-sided subshifts, the associated stabilized $C^*$-algebras are isomorphic so that their K-groups are isomorphic.

We study multifractal spectra of a simple class of interval transformations with a neutral fixed point. The analyticity of their spectral curves depends on whether the absolutely continuous invariant measure is infinite or not.

We show that every shift of finite type $X$ with positive entropy has proper subshifts of finite type with entropy strictly smaller than the entropy of $X$, but with entropy arbitrarily close to the entropy of $X$. Consequently, $X$ contains an infinite chain of subshifts of finite type which is strictly decreasing in entropy.

It has been conjectured that the stably ergodic diffeomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diffeomorphisms of a compact manifold. In this paper we deal with two recalcitrant examples; the standard map cross Anosov and the ergodic automorphisms of the 4-torus. In both cases we show that they may be approximated by stably ergodic diffeomorphisms which have the stable accessibility property.

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

A class of transformations on $[0,1]^2$, which includes transformations obtained by a Poincaré section of the Lorenz equation, is considered. We prove a formula which connects local dimension, entropy and characteristic exponents of ergodic invariant probability measures.

We give a simple proof for monotonicity of topological entropy in the quadratic family. We use a spectral property of a Ruelle operator.

Geometric problems are usually formulated by means of (exterior) differential systems. In this theory, one enriches the system by adding algebraic and differential constraints, and then looks for regular solutions. Here we adopt a dual approach, which consists of enriching a plane field, as this is often practised in control theory, by adding brackets of the vector fields tangent to it and, then, looking for singular solutions of the obtained distribution. We apply this to the isometry problem of rigid geometric structures.