scholarly journals SRB measures for almost Axiom A diffeomorphisms

2015 ◽  
Vol 36 (7) ◽  
pp. 2015-2043 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
RENAUD LEPLAIDEUR
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Compact Manifold ◽  
Invariant Set ◽  
Singular Set ◽  
Axiom A ◽  
Srb Measures ◽  
Hyperbolic Points

We consider a diffeomorphism $f$ of a compact manifold $M$ which is almost Axiom A, i.e. $f$ is hyperbolic in a neighborhood of some compact $f$-invariant set, except in some singular set of neutral points. We prove that if there exists some $f$-invariant set of hyperbolic points with positive unstable Lebesgue measure such that for every point in this set the stable and unstable leaves are ‘long enough’, then $f$ admits an SRB (probability) measure.

scholarly journals Capacity of attractors

1981 ◽  
Vol 1 (3) ◽  
pp. 381-388 ◽  
Author(s):  
Lai-Sang Young
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Lyapunov Exponents ◽  
Borel Probability Measure ◽  
Invariant Set ◽  
Axiom A ◽  
Borel Probability

AbstractLet f be a diffeomorphism of a manifold and Λ be an f-invariant set supporting an ergodic Borel probability measure μ with certain properties. A lower bound on the capacity of Λ is given in terms of the μ-Lyapunov exponents. This applies in particular to Axiom A attractors and their Bowen-Ruelle measure.

scholarly journals SRB measures for partially hyperbolic attractors of local diffeomorphisms

2018 ◽  
Vol 40 (6) ◽  
pp. 1545-1593
Author(s):  
ANDERSON CRUZ ◽  
PAULO VARANDAS
Keyword(s):  
Thermodynamic Formalism ◽  
Positive Lebesgue Measure ◽  
Stable Bundle ◽  
Local Diffeomorphism ◽  
Local Diffeomorphisms ◽  
Axiom A ◽  
Srb Measures ◽  
Cone Field ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

Multifractal formalism for the inverse of random weak Gibbs measures

2019 ◽  
Vol 20 (04) ◽  
pp. 2050024
Author(s):  
Zhihui Yuan
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Gibbs Measures ◽  
Borel Probability Measure ◽  
Cantor Set ◽  
Multifractal Formalism ◽  
The Real ◽  
Random Dynamics ◽  
Borel Probability

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

scholarly journals Bistable vector fields are axiom A

1995 ◽  
Vol 51 (1) ◽  
pp. 83-86
Author(s):  
Mike Hurley
Keyword(s):  
Vector Field ◽  
Compact Manifold ◽  
Vector Fields ◽  
Axiom A ◽  
Smooth Compact Manifold ◽  
Structurally Stable

Recently L. Wen showed that if a C1 vector field (on a smooth compact manifold without boundary) is both structurally stable and topologically stable then it will satisfy Axiom A. The purpose of this note is to indicate how results from an earlier paper can be used to simplify somewhat Wen's argument.

scholarly journals CANARD CYCLES IN GLOBAL DYNAMICS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250026 ◽  
Author(s):  
ALEXANDRE VIDAL ◽  
JEAN-PIERRE FRANÇOISE
Keyword(s):  
Invariant Set ◽  
Invariant Sets ◽  
Global Dynamics ◽  
Relaxation Oscillations ◽  
Singular Set ◽  
Connected Component ◽  
Slow Dynamics ◽  
Fast Recovery ◽  
Transition Points ◽  
Stable Invariant

Fast-slow systems are studied usually by "geometrical dissection" [Borisyuk & Rinzel, 2005]. The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen as a parameter of the fast dynamics. A generic solution comes close to a connected component of the stable invariant sets of the fast dynamics. As the slow dynamics evolves, this attractor may lose its stability and the solution eventually reaches quickly another connected component of attractors of the fast dynamics and the process may repeat. This scenario explains quite well relaxation oscillations and more complicated oscillations like bursting. More recently, in relation both with theory of dynamical systems [Dumortier & Roussarie, 1996] and with applications to physiology [Desroches et al., 2008; Toporikova et al., 2008], a new interest has emerged in canard cycles. These orbits share the property that they remain for a while close to an unstable invariant set (either singular set or periodic orbits of the fast dynamics). Although canards were first discovered when the transition points were folds, in this article, we focus on the case where one or several transition points (or "jumps") are instead transcritical. We present several new surprising effects like the "amplification of canards" or the "exceptionally fast recovery" on both (1 + 1)-systems and (2 + 1)-systems associated with tritrophic food chain dynamics. Finally, we also mention their possible relevance to the notion of resilience which has been coined out in ecology [Holling, 1973; Ludwig et al., 1997; Martin, 2004].

scholarly journals SRB measures for Axiom A endomorphisms

2004 ◽  
Vol 11 (6) ◽  
pp. 785-797 ◽  
Author(s):  
Mariusz Urbanski ◽  
Christian Wolf
Keyword(s):  
Axiom A ◽  
Srb Measures

scholarly journals Ω-Stability is not dense in Axiom A

1988 ◽  
Vol 8 (4) ◽  
pp. 621-632
Author(s):  
S. E. Patterson
Keyword(s):  
Compact Manifold ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Axiom A

AbstractAn example of a diffeomorphism f on a three dimensional manifold M3 is constructed so that f satisfies Axiom A, has a cycle and f has a neighborhood N in Diffr (M3) so that each g in N is not Ω-stable. Existing techniques allow one to extend this example to any compact manifold of dimension greater than two.

scholarly journals Topological and symbolic dynamics for hyperbolic systems with holes

2010 ◽  
Vol 31 (5) ◽  
pp. 1305-1323 ◽  
Author(s):  
STEFAN BUNDFUSS ◽  
TYLL KRÜGER ◽  
SERGE TROUBETZKOY
Keyword(s):  
Finite Type ◽  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Symbolic Representation ◽  
Invariant Set ◽  
Subshift Of Finite Type ◽  
Axiom A ◽  
Topologically Transitive ◽  
Markov Map

AbstractWe consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Ω* of orbits which never falls into a fixed hole. We study various aspects of the symbolic representation of Ω* and of its non-wandering set Ωnw. Our results are on the cardinality of the set of topologically transitive components of Ωnw and their structure. We also prove that Ω* is generically a subshift of finite type in several senses.

Bounded Geometry in the Supports of Ergodic Invariant Probability Measures

1998 ◽  
Vol 08 (10) ◽  
pp. 1957-1973 ◽  
Author(s):  
Jun Hu
Keyword(s):  
Hausdorff Dimension ◽  
Periodic Orbit ◽  
Probability Measure ◽  
Lebesgue Measure ◽  
Turning Points ◽  
Invariant Probability Measure ◽  
Cantor Set ◽  
Probability Measures ◽  
Bounded Geometry

In this paper we put some techniques and methods of [Misurewicz, 1979; Hu & Sullivan, 1997; Hu & Tresser, 1998; Blokh & Lyubich, 1990; Martens et al., 1992] together to show that if a map f, from an interval I into itself with finitely many turning points, satisfies a new smooth regularity in [Hu & Sullivan, 1997] and is on the boundary of chaos, and if μ is an ergodic f-invariant probability measure on I which is not concentrated on a periodic orbit of f, then the support K of μ is a Cantor set of bounded geometry, and hence has Lebesgue measure 0 and Hausdorff dimension strictly between 0 and 1. We also include some natural examples which satisfy this new smooth regularity rather than the traditional ones.

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