For projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 that imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.

We prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

We consider a class of Gibbs measures on self-affine Sierpiński carpets and perform the multifractal analysis of its elements. These deterministic measures are Gibbs measures associated with bundle random dynamical systems defined on probability spaces whose geometrical structure plays a central role. A special subclass of these measures is the class of multinomial measures on Sierpiński carpets. Our result improves the already known result concerning the multifractal nature of the elements of this subclass by considerably weakening and in some cases even eliminating a strong separation condition of geometrical nature.

We prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point p∈M that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. p∈M that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.

We show that, for C1-generic diffeomorphisms, every chain recurrent class C that has a partially hyperbolic splitting with dimEc=1 either is an isolated hyperbolic periodic orbit, or is accumulated by non-trivial homoclinic classes. We also prove that, for C1-generic diffeomorphisms, any chain recurrent class that has a dominated splitting with dim(E)=1 either is a homoclinic class, or the bundle E is uniformly contracting. As a corollary we prove in dimension three a conjecture of Palis, which announces that any C1-generic diffeomorphism is either Morse–Smale, or has a non-trivial homoclinic class.

Averages of functionals along trajectories are studied by evaluating the averages along chains. This yields results for the possible limits and, in particular, for ergodic limits. Applications to Lyapunov exponents and to concepts of rotation numbers of linear Hamiltonian flows and of general linear flows are given.

Our goal in this paper is to give an example of a one-parameter family of rational maps for which, in the parameter plane, there is a Cantor set of simple closed curves consisting of parameters for which the corresponding Julia set is a Sierpiński curve. Hence, the Julia sets for each of these parameters are homeomorphic. However, each of the maps in this set is dynamically distinct from (i.e. not topologically conjugate to) any other map in this set (with only finitely many exceptions). We also show that, in the dynamical plane for any map drawn from a large open set in the connectednesslocus in this family, there is a Cantor set of invariant simple closed curves on which the map is conjugate to the product of certain subshifts of finite type with the maps on the unit circle.

We introduce a new tool to study the spectral type of rank-one transformations using the method of the central limit theorem for trigonometric sums. We get some new applications.

We prove that if is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of which leaves invariant a compact set then there is a common fixed point for all elements of . We also show that if is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of with index at most two.

We find an explicit formula for the invariant density of a generalized β-map. This allows us to also find an explicit formula for the invariant density of Chebyshev map and discuss the monotonicity of the asymptotic average for such maps. Our results are based on a generalization of works of Parry (Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416; 15 (1964), 95–105).

We characterize when C0-semigroups induced by semiflows are hypercyclic, topologically mixing, or chaotic both on spaces of integrable functions and on spaces of continuous functions. Furthermore, we give characterizations of transitivity for weighted composition operators on these spaces.

We generalize a series of topological Wiener–Wintner ergodic theorems due to Walters to the context of group extensions of measure-preserving transformations where the group is a non-abelian compact Lie group. Applications to random ergodic theorems for a shift map are given.

Random diffeomorphisms with bounded absolutely continuous noise are known to possess a finite number of stationary measures. We discuss the dependence of stationary measures on an auxiliary parameter, thus describing bifurcations of families of random diffeomorphisms. A bifurcation theory is developed under mild regularity assumptions on the diffeomorphisms and the noise distribution (e.g. smooth diffeomorphisms with uniformly distributed additive noise are included). We distinguish bifurcations where the density function of a stationary measure varies discontinuously or where the support of a stationary measure varies discontinuously. We establish that generic random diffeomorphisms are stable. The densities of stable stationary measures are shown to be smooth and to depend smoothly on an auxiliary parameter, except at bifurcation values. The bifurcation theory explains the occurrence of transients and intermittency as the main bifurcation phenomena in random diffeomorphisms. Quantitative descriptions by means of average escape times from sets as functions of the parameter are provided. Further quantitative properties are described through the speed of decay of correlations as a function of the parameter. Random differentiable maps which are not necessarily injective are studied in one dimension; we show that stable one-dimensional random maps occur open and dense and that in one-parameter families bifurcations are typically isolated. We classify codimension-one bifurcations for one-dimensional random maps; we distinguish three possible kinds, the random saddle node, the random homoclinic and the random boundary bifurcation. The theory is illustrated on families of random circle diffeomorphisms and random unimodal maps.