In this paper we study the dynamics of a rational function \phi\in K(z) defined over some finite extension K of \mathbb{Q}_p. After proving some basic results, we define a notion of ‘components’ of the Fatou set, analogous to the topological components of a complex Fatou set. We define hyperbolic p-adic maps and, in our main theorem, characterize hyperbolicity by the location of the critical set. We use this theorem and our notion of components to state and prove an analogue of Sullivan's No Wandering Domains Theorem for hyperbolic maps.

We define a normalized entropy functional for compact Finsler manifolds of negative flag curvature. Using the method of Besson, Courtois, and Gallot, we show that among all such manifolds that are homotopy equivalent to a compact, Riemannian, locally symmetric manifold of negative curvature, the entropy functional is minimized precisely on the locally symmetric manifold.

We study \mathcal{C}^k-diffeomorphisms, k\ge 1, f: M\to M, exhibiting heterodimensional cycles (i.e. cycles containing periodic points of different stable indices). We prove that if f can not be \mathcal{C}^k-approximated by diffeomorphisms with homoclinic tangencies, then f is in the closure of an open set \mathcal{U}\subset \operatorname{Diff}^k(M) consisting of diffeomorphisms g with a non-hyperbolic transitive set \Lambda_g which is locally maximal and strongly partially hyperbolic (the partially hyperbolic splitting at \Lambda_g has three non-trivial directions). As a consequence, in the case of 3-manifolds, we give new examples of open sets of \mathcal{C}^1-diffeomorphisms for which residually infinitely many sinks or sources coexist (\mathcal{C}^1-Newhouse's phenomenon). We also prove that the occurrence of non-hyperbolic dynamics has persistent character in the unfolding of heterodimensional cycles.

A homeomorphism on a metric space (X,d) is completely scrambled if for each x\not= y\in X, \lim sup_{n\longrightarrow +\infty} d(f^n(x),f^n(y))>0 and \lim inf_{n\ longrightarrow +\infty}d(f^n(x),f^n(y))=0. We study the basic properties of completely scrambled homeomorphisms on compacta and show that there are ‘many’ compacta admitting completely scrambled homeomorphisms, which include some countable compacta (we give a characterization), the Cantor set and continua of arbitrary dimension.

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

Pointwise convergence a.e. of \mu^k f has been studied by Bellow, Jones and Rosenblatt. We answer two of their questions: the strong sweeping out property holds (in this case) iff the measure \mu has unbounded angular ratio and the stability is equivalent to the bounded angular ratio.

Under some weak hyperbolicity conditions, we establish C^0- and C^\infty-local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0-local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0-local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty-local rigidity is deduced from the C^0-local rigidity following a procedure outlined by Katok and Spatzier.

Using similar considerations, we also establish C^0-global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

We estimate the decay of correlations for some Markov maps on a countable state space. A necessary and sufficient condition is given for the transfer operator to be quasi-compact on the space of locally Lipschitz functions. In the non-quasi-compact case, the decay of correlations depends on the contribution to the transfer operator of the complementary of finitely many cylinders. Estimates are given for some non-uniformly expanding maps and for maps with bounded jumps.

For a piecewise monotonic map T:X\to{\Bbb R}, where X is a finite union of closed intervals, define R(T)= \bigcap_{n=0}^{\infty}\overline{T^{-n}X}. The influence of small perturbations of T on the dynamical system (R(T),T) is investigated. If P is a finite and T-invariant subset of R(T), and if f_0:P\to{\Bbb R} is a non-negative continuous function, then it is proved that the infimum of the topological pressure p(R(T),T,f) over all non-negative continuous functions f:X\to{\Bbb R} with f|_P=f_0 equals the maximum of h_{\text{\rm top}}(R(T),T) and p(P,T,f_0). This result is used to obtain stability conditions, which are equivalent to the upper semi-continuity of the topological pressure for every continuous function f:X\to{\Bbb R}. In the case of a continuous piecewise monotonic map T:X\to{\Bbb R} one of these stability conditions is: there exists no endpoint of an interval of monotonicity of T which is periodic and contained in the interior of X. Furthermore, these results are applied to monotonic mod one transformations, another special case of piecewise monotonic maps.

On considère l'espace {\mathcal B}_2 des homéomorphismes du plan obtenus en recollant deux translations, muni de la topologie compacte-ouverte. On construit une famille dans {\mathcal B}_2, continûment indexée par les éléments du Cantor \{0,1\}^\mathbb{Z}, pour laquelle les classes de conjugaison correspondent aux orbites du décalage (ou “ shift ”) sur \{0,1\}^\mathbb{Z}. Ceci permet de montrer qu'il n'y a pas de classification borélienne de la relation de conjugaison sur {\mathcal B}_2.

Given a two-parameter family of three-dimensional diffeomorphisms \{f_{a,b}\}_{a,b}, with a dissipative (but not sectionally dissipative) saddle fixed point, assume that a special type of quadratic homoclinic tangency of the invariant manifolds exists. Then there is a return map f^n_{a,b}, near the homoclinic orbit, for values of the parameter near such a tangency and for n large enough, such that, after a change of variables and reparametrization depending on n, this return map tends to a simple quadratic map. This implies the existence of strange attractors and infinitely many sinks as in other known cases. Moreover, there appear attracting invariant circles, implying the existence of quasi-periodic behaviour near the homoclinic tangency.

We prove the following property: if X in \mathcal G^1(M) has no singularity and x \in \Sigma(X), then \overline{\operatorname{orbit}(x)} \cap \overline{\operatorname{per}(X)} \not = \emptyset. In addition, if we assume \overline{\operatorname{per}_i(X)} \cap \overline{\operatorname{per}_j(X)} = \emptyset for i \not = j, then \overline{\operatorname{per}(X)} = \bigcup_{i=0}^{n-1} \overline{\operatorname{per}_i(X)} is a hyperbolic set. Moreover, we shall give a proof of the \Omega-stability conjecture for flows.