Collision, explosion and collapse of homoclinic classes

Díaz, Lorenzo J.; Santoro, Bianca

doi:10.1088/0951-7715/17/3/013

Cited by 6 publications

(6 citation statements)

References 24 publications

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“…Regarding the dynamics at the collision parameter t = t c , we recall that collisions of homoclinic classes were studied from the merely topological point of view in [31,30]. We also observe an IFS somewhat similar to the one associated to the maps f0 , f1,tc considered here also appears in [32,1,2], where different questions were being asked.…”

Section: 4supporting

confidence: 55%

“…See Remark 11.4 for a complete description. Finally, the parameter t c can be also seen as a parameter of collision between Γ (tc) and another Ftc -maximal invariant set from outside Σ 2 × [0, 1], leading to a collision of homoclinic classes in the spirit of [31,30]. This gives rise to explosions of topological entropy and of the space of invariant measures of Ftc | Γ (tc) , see Propositions 11.6 and 11.7.…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…The skew-products lead to globally defined one-parameter families that may have "explosions of entropy and of the space of ergodic measures" at some bifurcation associated to a collision of a pair of homoclinic classes. This sort of question was studied in [31,30], where the collision occurs through the orbit of a parabolic (saddle-node) point. In our setting the collision may be of different nature.…”

Section: Lemma 918 (Uniform Contractions) the Ifs {Gmentioning

confidence: 99%

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

Díaz

Gelfert

Rams

2022

DCDS

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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.

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Section: 4supporting

confidence: 55%

Section: Summary Of Resultsmentioning

confidence: 99%

Section: Lemma 918 (Uniform Contractions) the Ifs {Gmentioning

confidence: 99%

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

Díaz

Gelfert

Rams

2022

DCDS

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“…10. Lower semicontinuity should not hold in general: e.g., different homoclinic classes of maximal entropy may sometimes merge through an arbitrarily small perturbation, see [43] for an example in dimension 3. These examples also indicate that the limit of ergodic m.m.e.…”

Section: Borel Classificationmentioning

confidence: 99%

Measures of maximal entropy for surface diffeomorphisms

Buzzi¹,

Crovisier²,

Sarig³

2018

Preprint

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We show that C ∞ surface diffeomorphisms with positive topological entropy have at most finitely many ergodic measures of maximal entropy in general, and at most one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irreducible countable state Markov shifts.

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“…diffeomorphisms in a residual subset of Diff 1 (M)) non-disjoint homoclinic classes coincide, see [5,12]. For examples of overlapping homoclinic classes (each class is not contained in the other one and the classes have non-empty intersection) see [25].…”

Section: Introductionmentioning

confidence: 99%