Skew product cycles with rich dynamics: From totally non-hyperbolic dynamics to fully prevalent hyperbolicity

Díaz, Lorenzo J.; Esteves, Salete; Rocha, Jorge

doi:10.1080/14689367.2015.1085492

Cited by 8 publications

(7 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Observe that our definitions do not involved any transversality assumption (indeed in our context of a skew product such a transversality does not make sense, see also [6,Section 3] for more details on homoclinic relations for skew products). However, due to the fact that the maps f 0 and f 1 have no critical points, the homoclinic points behave as the transverse ones in the differentiable setting.…”

Section: Appendix a Transitivity And Homoclinic Relations Proof Of Pr...mentioning

confidence: 99%

The structure of the space of ergodic measures of transitive partially hyperbolic sets

et al. 2019

View full text Add to dashboard Cite

We provide examples of transitive partially hyperbolic dynamics (specific but paradigmatic examples of homoclinic classes) which blend different types of hyperbolicity in the one-dimensional center direction. These homoclinic classes have two disjoint parts: an "exposed" piece which is poorly homoclinically related with the rest and a "core" with rich homoclinic relations. There is an associated natural division of the space of ergodic measures which are either supported on the exposed piece or on the core. We describe the topology of these two parts and show that they glue along nonhyperbolic measures.Measures of maximal entropy are discussed in more detail. We present examples where the measure of maximal entropy is nonhyperbolic. We also present examples where the measure of maximal entropy is unique and nonhyperbolic, however in this case the dynamics is nontransitive.

show abstract

Section: Appendix a Transitivity And Homoclinic Relations Proof Of Pr...mentioning

confidence: 99%

The structure of the space of ergodic measures of transitive partially hyperbolic sets

et al. 2019

View full text Add to dashboard Cite

show abstract

“…The following Lemma, whose proof is omitted here (for details [DER,Propositions 7.5 and 7.10]), assures that all points on Λ t \ {P, Q} have some iterate in ∆ t and associates an itinerary to each point of ∆ t having returns to ∆ t .…”

Section: Dynamics In a Neighborhood Of The Cycle Prove The Main Resultsmentioning

confidence: 99%

Growth of number of periodic orbits of one family of skew product maps

Esteves

2017

Dynamical Systems

Self Cite

View full text Add to dashboard Cite

show abstract

“…Proof of Proposition 5.1. To continue our discussion, we state a simple lemma about homoclinic relations which is just a reformulation of [20,Corollary 3.1] for the fixed points P and Q of F . For completeness and to illustrate the dynamics, we will sketch its proof.…”

Section: 2mentioning

confidence: 99%

“…The constructions in [20] provide an explicit two-parameter family (with parameters a and t) of fiber maps f 0 = g a (concave) and f 1 = g 1,t (affine) such that the corresponding skew-product map Fa,t falls into one of the following cases, according to the choices of the parameters a and t:…”

Section: Illustration Of the Parabolic Case Ii)mentioning

confidence: 99%

Mingled hyperbolicities: Ergodic properties and bifurcation phenomena (an approach using concavity)

Díaz

Gelfert

Rams

2022

DCDS

Self Cite

View full text Add to dashboard Cite

We consider skew-products with concave interval fiber maps over a certain subshift obtained as the projection of orbits staying in a given region. It generates a new type of (essentially) coded shift. The fiber maps have expanding and contracting regions which dynamically interact. The dynamics also exhibits pairs of horseshoes of different type of hyperbolicity which, in some cases, are cyclically related.The space of ergodic measures on the base is an entropy-dense Poulsen simplex. Those measures lift canonically to ergodic measures for the skew-product. We explain when and how the spaces of (fiber) contracting and expanding ergodic measures glue along the nonhyperbolic ones. A key step is the approximation (in the weak<inline-formula><tex-math id="M1">\begin{document}$ \ast $\end{document}</tex-math></inline-formula> topology and in entropy) of nonhyperbolic measures by ergodic ones, obtained only by means of concavity. Concavity is not merely a technical artificial hypothesis, but it prevents the presence of additional independent subsystems. The description of homoclinic relations is also a key instrument.These skew-products are embedded in non-decreasing entropy one-parameter family of diffeomorphisms stretching from a heterodimensional cycle to a collision of homoclinic classes. Associated bifurcation phenomena involve a jump of the space of ergodic measures and, in some cases, of entropy.

show abstract

Skew product cycles with rich dynamics: From totally non-hyperbolic dynamics to fully prevalent hyperbolicity

Cited by 8 publications

References 17 publications

The structure of the space of ergodic measures of transitive partially hyperbolic sets

The structure of the space of ergodic measures of transitive partially hyperbolic sets

Growth of number of periodic orbits of one family of skew product maps

Mingled hyperbolicities: Ergodic properties and bifurcation phenomena (an approach using concavity)

Contact Info

Product

Resources

About