22,915 research outputs found
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 the set of points with the same itinerary of 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 . We shall prove that {\em all} singular-hyperbolic
attractors on compact 3-manifolds have topological dimension . The
proof uses the methods in \cite{MP}.Comment: 18 pages, 1 figur
Characterizing finite sets of nonwandering points
We characterize finite sets of nonwandering points for generic
diffeomorphisms as those which are {\em uniformly bounded}, i.e., there is
an uniform bound for small perturbations of the derivative of along the
points in up to suitable iterates. We use this result to give a
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 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 in an isolating block of a
singular-hyperbolic attractor for three-dimensional vector fields . We prove
that for every vector field close to the set
contains a singularity is {\em residual} in . 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
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