Existence of attractors for three-dimensional flows

We prove the results in [1] using Theorem 1 of the recent paper [2] by Crovisier and Yang. References: [1] Arbieto, A., Rojas, C., Santiago, B., Existence of attractors, homoclinic tangencies and singular-hyperbolicity for flows, arXiv:1308.1734v1 [math.DS] 8 Aug 2013. [2] Crovisier, S., Yang, D., On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis, arXiv:1404.5130v1 [math.DS] 21 Apr 2014.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1308.1734 by other author

On supports of expansive measures

We prove that a homeomorphism of a compact metric space has an expansive measure \cite{ms} if and only if it has many ones with invariant support. We also study homeomorphisms for which the expansive measures are dense in the space of Borel probability measures. It is proved that these homeomorphisms exhibit a dense set of Borel probability measures which are expansive with full support. Therefore, their sets of heteroclinic points has no interior and the spaces supporting them have no isolated points.Comment: 7 page

On pairwise sensitive homeomorphisms

We obtain properties of the pairwise sensitive homeomorphisms defined in \cite{cj}. For instance, we prove that their sets of points with converging semi-orbits have measure zero, that such homeomorphisms do not exist in a compact interval and, in the circle, they are the Denjoy ones. Applications including alternative proofs of well-known facts in expansive systems are given.Comment: 9 page

Partition's sensitivity for measurable maps

We study countable partitions for measurable maps on measure spaces such that for all point $x$ the set of points with the same itinerary of $x$ is negligible. We prove that in nonatomic probability spaces every strong generator (Parry, W., {\em Aperiodic transformations and generators}, J. London Math. Soc. 43 (1968), 191--194) satisfies this property but not conversely. In addition, measurable maps carrying partitions with this property are aperiodic and their corresponding spaces are nonatomic. From this we obtain a characterization of nonsingular countable to one mappings with these partitions on nonatomic Lebesgue probability spaces as those having strong generators. Furthermore, maps carrying these partitions include the ergodic measure-preserving ones with positive entropy on probability spaces (thus extending a result in Cadre, B., Jacob, P., {\em On pairwise sensitivity}, J. Math. Anal. Appl. 309 (2005), no. 1, 375--382). Some applications are given.Comment: 13 page

Topological dimension of singular-hyperbolic attractors

An {\em attractor} is a transitive set of a flow to which all positive orbit close to it converges. An attractor is {\em singular-hyperbolic} if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction \cite{MPP}. The geometric Lorenz attractor \cite{GW} is an example of a singular-hyperbolic attractor with topological dimension $\geq 2$. We shall prove that {\em all} singular-hyperbolic attractors on compact 3-manifolds have topological dimension $\geq 2$. The proof uses the methods in \cite{MP}.Comment: 18 pages, 1 figur

Characterizing finite sets of nonwandering points

We characterize finite sets $S$ of nonwandering points for generic diffeomorphisms $f$ as those which are {\em uniformly bounded}, i.e., there is an uniform bound for small perturbations of the derivative of $f$ along the points in $S$ up to suitable iterates. We use this result to give a $C^1$ generic characterization of the Morse-Smale diffeomorphisms related to the weak Palis conjecture \cite{c}. Furthermore, we obtain another proof of the result by Liao and Pliss about the finiteness of sinks and sources for star diffeomorphisms \cite{l}, \cite{Pl}.Comment: 17 page

Equicontinuity on semi-locally connected spaces

We show that a homeomorphism of a semi-locally connected compact metric space is equicontinuous if and only if the distance between the iterates of a given point and a given subcontinuum (not containing that point) is bounded away from zero. This is false for general compact metric spaces. Moreover, homeomorphisms for which the conclusion of this result holds satisfy that the set of automorphic points contains those points where the space is not semi-locally connected.Comment: 6 page

Entropy, pseudo-orbit tracing property and positively expansive measures

We study homeomorphisms of compact metric spaces whose restriction to the nonwandering set has the pseudo-orbit tracing property. We prove that if there are positively expansive measures, then the topological entropy is positive. Some short applications of this result are included.Comment: 6 page

Some properties of surface diffeomorphisms

We obtain some properties of $C^1$ generic surface diffeomorphisms as finiteness of {\em non-trivial} attractors, approximation by diffeomorphisms with only a finite number of {\em hyperbolic} homoclinic classes, equivalence between essential hyperbolicity and the hyperbolicity of all {\em dissipative} homoclinic classes (and the finiteness of spiral sinks). In particular, we obtain the equivalence between finiteness of sinks and finiteness of spiral sinks, abscence of domination in the set of accumulation points of the sinks, and the equivalence between Axiom A and the hyperbolicity of all homoclinic classes. These results improve \cite{A}, \cite{a}, \cite{m} and settle a conjecture by Abdenur, Bonatti, Crovisier and D\'{i}az \cite{abcd}.Comment: 28 page

Omega-limit sets close to singular-hyperbolic attractors

We study the omega-limit sets $\omega_X(x)$ in an isolating block $U$ of a singular-hyperbolic attractor for three-dimensional vector fields $X$. We prove that for every vector field $Y$ close to $X$ the set $\{x\in U:\omega_Y(x)$ contains a singularity$\}$ is {\em residual} in $U$. This is used to prove the persistence of singular-hyperbolic attractors with only one singularity as chain-transitive Lyapunov stable sets. These results generalize well known properties of the geometric Lorenz attractor \cite{gw} and the example in \cite{mpu}.Comment: 17 page