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We consider the space of C1-diffeomorphims of a three dimensional closed manifold equipped with the C1-topology. It is known that there are open sets in which C1-generic diffeomorphisms display uncountably many chain recurrence classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of this paper. The aim of the paper is to show that aperiodic classes of C1-generic diffeomorphisms can exhibit a variety of topological properties. More specifically, there are C1-generic diffeomorphisms with (1) minimal expansive aperiodic classes, (2) minimal but non-uniquely ergodic aperiodic classes, (3) transitive but non-minimal aperiodic classes, (4) non-transitive, uniquely ergodic aperiodic classes.

We consider a class of C4 partially hyperbolic systems on T2 described by maps Fε(x,θ)=(f(x,θ),θ+εω(x,θ)) where f(·,θ) are expanding maps of the circle. For sufficiently small ε and ω generic in an open set, we precisely classify the SRB measures for Fε and their statistical properties, including exponential decay of correlation for Hölder observables with explicit and nearly optimal bounds on the decay rate.

Given a continuous linear cocycle$\\mathcal {A}$over a homeomorphism f of a compact metric space X , we investigate its set$\\mathcal {R}$of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set$\\mathcal {R}$is of first Baire category (i.e., meager) in X , unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

We prove that generic fiber-bunched and Hölder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a $u$-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched linear cocycles.

We prove that a partially hyperbolic attractor for a C 1 vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is C 2 , and the center bundle admits the sectionally expanding condition w.r.t. Gibbs u -states, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.

In this paper we consider C 1 surface diffeomorphisms and study the existence of phase transitions, here expressed by the non-analiticity of the pressure function associated to smooth and geometric-type potentials. We prove that the space of C 1 -surface diffeomorphisms admitting phase transitions is a C 1 -Baire generic subset of the space of non-Anosov diffeomorphisms. In particular, if S is a compact surface which is not homeomorphic to the 2-torus then a C 1 -generic diffeomorphism on S has phase transitions. We obtain similar statements in the context of C 1 -volume preserving diffeomorphisms. Finally, we prove that a C 2 -surface diffeomorphism exhibiting a dominated splitting admits phase transitions if and only if has some non-hyperbolic periodic point.

We consider skew products on $M\\times \\mathbb{T}^{2}$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $\\unicode[STIX]{x1D6FA}$-non-stable systems introduced by Abraham and Smale [Non-genericity of Ł-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5–8.] and Shub [Topological Transitive Diffeomorphisms in$T^{4}$ (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.

Let X be a C 1+ vector field on a compact Riemannian manifold M with dimension d ≥ 3. Let Λ be a transitive singular hypebolic set with positive volume. We show that Λ = M and Λ is a uniformly hyperbolic set without singularities.

Let $g:M\\rightarrow M$ be a $C^{1+\\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on $M$. We show that, if $f:M\\rightarrow M$ is a $C^{1+\\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of $f$ and $g$ span the whole tangent space at some point on $M$, the set of points that equidistribute under $g$ but have non-dense orbits under $f$ has full Hausdorff dimension. The same result is also obtained when $M$ is the torus and $f$ is a toral endomorphism whose center-stable subspace does not contain the stable subspace of $g$ at some point.

We give a brief survey on the dynamics of vector fields with singularities. The aim of this survey is not to list all results in this field, but only to introduce some results from several viewpoints and some technics.