Hyperbolic Basic Set Research Articles

Regularity of the Hausdorff dimension of some hyperbolic systems

We study the regularity of the Hausdorff dimension for certain hyperbolic basic sets, as a function of the dynamical system. The main results ensure that the Hausdorff dimension of certain multidimensional thin hyperbolic attractors and of certain nonconformal iterated function systems varies smoothly and analytically with respect to the diffeomorphism. Similar results hold for average conformal hyperbolic sets and average conformal repellers.

Almost blenders and parablenders

AbstractA blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds robustly contains an open set. Introduced by Bonatti and Díaz in the 1990s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper we introduce analogous notions in a measurable setting. We define an almost blender as a hyperbolic basic set for which a prevalent perturbation has a local unstable set having positive Lebesgue measure. Almost parablenders are defined similarly in terms of jets. We study families of endomorphisms of $\mathbb {R}^2$ leaving invariant the continuation of a hyperbolic basic set. When an inequality involving the entropy and the maximal contraction along stable manifolds is satisfied, we obtain an almost blender or parablender. This answers partially a conjecture of Berger, and complements previous works on the construction of blenders by Avila, Crovisier, and Wilkinson or by Moreira and Silva. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak, and Urbański, we study families of skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.

Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension

In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) a hyperbolic basic set which exhibits $C^2$-robust homoclinic tangency of the largest codimension for any higher dimensional manifold, using blenders. The former implies that an analog of Moreira's theorem on Cantor sets in the real line does not hold in higher dimension. The latter solves a question posed by Barrientos and A.Raibekas.

On local structure of one-dimensional basic sets of non-reversible A-endomorphisms of surfaces

Recently the authors of the article discovered a meaningful class of non-reversible endomorphisms on a two-dimensional torus. A remarkable property of these endomorphisms is that their non-wandering sets contain nontrivial one-dimensional strictly invariant hyperbolic basic sets (in the terminology of S. Smale and F. Pshetitsky) which have the uniqueness of an unstable one-dimensional bundle. It was proved that nontrivial (other than periodic isolated orbits) invariant sets can only be repellers. Note that this is not the case for reversible endomorphisms (diffeomorphisms). In the present paper, it is proved that one-dimensional expanding uniquely hyperbolic and strictly invariant one-dimensional expanding attractors and one-dimensional contracting repellers of non-reversible A-endomorphisms of closed orientable surfaces have the local structure of the product of an interval by a zero-dimensional closed set (finite or Cantor). This result contrasts with the existence of one-dimensional fractal repellers arising in complex dynamics on the Riemannian sphere and not possessing the properties of the existence of a single one-dimensional unstable bundle.

Uniform hyperbolicity revisited: index of periodic points and equidimensional cycles

ABSTRACTIn this paper, we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r ≥ 1). In the case of the C1-topology, we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum C1-robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of f are continuous in the weak*-topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The latter can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures.

The centralizer of $C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial

The centralizer of $C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial

Expansive transitive sets for robust and generic diffeomorphisms

ABSTRACTLet f be a diffeomorphism on a compact smooth Riemannian manifold M, and let Λ be a closed f-invariant transitive subset of M. In this paper, we show that f|Λ is C1-stably expansive if and only if Λ is a hyperbolic basic set. Moreover, C1-generically, a locally maximal transitive set Λ is expansive if and only if it is a hyperbolic basic set.

HYPERBOLICITY OF CHAIN TRANSITIVE SETS WITH LIMIT SHADOWING

Abstract. In this paper we show that any chain transitive set of a dif-feomorphism on a compact C ∞ -manifold which is C 1 -stably limit shad-owable is hyperbolic. Moreover, it is proved that a locally maximal chaintransitive set of a C 1 -generic diffeomorphism is hyperbolic if and only ifit is limit shadowable. Transitivesets, homoclinicclassesand chaincomponentsofadiffeomorphismare natural candidates to replace the hyperbolic basic sets in nonhyperbolictheory of differentiable dynamical systems, and many recent papers exploredtheir “hyperbolic-like” properties (for more details, see [2, 6, 8, 9, 14, 15]).In this paper we study the chain transitive sets which are limit shadowable.Let us be more precise. Let M be a compact C ∞ -manifold, and let Diff(M) bethe space of diffeomorphisms of M endowed with the C 1 -topology. Denote byd the distance on M induced from a Riemannian metric on the tangent bundleTM. For δ > 0, a sequence {x n } n∈Z in Λ is called a δ-limit chain iflim |n|→∞ d(f(x

The ergodic shadowing property from the robust and generic view point

In this paper, we discuss that if a diffeomorphisms has the -stably ergodic shadowing property in a closed set, then it is a hyperbolic elementary set. Moreover, -generically: if a diffeomorphism has the ergodic shadowing property in a locally maximal closed set, then it is a hyperbolic basic set. MSC:34D30, 37C20.

Chain components with stably limit shadowing property are hyperbolic

Let f be a diffeomorphism on a closed smooth manifold M. In this paper, we show that f has the -stably limit shadowing property on the chain component of f containing a hyperbolic periodic point p, if and only if is a hyperbolic basic set. MSC:37C50, 34D10.

Shadowable chain transitive sets

Let f be a diffeomorphism of a closed manifold M, and let be a closed f-invariant set. In this paper, by applying the reasoning developed in Wen et al. [J. Differ. Equ. 246 (2009), pp. 340–357] based on Liao, we study chain transitive sets in view of shadowing theory, and it is proved that is chain transitive and -stably shadowing if and only if is hyperbolic basic set. As a corollary, for a chain component of f, it is proved that is -stably shadowing if and only if is a hyperbolic homoclinic class.

Equilibrium measures on saddle sets of holomorphic maps on $$\mathbb{P }^2$$

We consider the case of hyperbolic basic sets \(\Lambda \) of saddle type for holomorphic maps \(f:{\mathbb{P }}^2{\mathbb{C }}\rightarrow {\mathbb{P }}^2{\mathbb{C }}\). We study equilibrium measures \(\mu _\phi \) associated to a class of Holder potentials \(\phi \) on \(\Lambda \), and find the measures \(\mu _\phi \) of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension \(\delta _{\mu _\phi }\) of \(\mu _\phi \) that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of \(\mu _\phi \) in the case when the preimage counting function is constant on \(\Lambda \). For terminal/minimal saddle sets we prove that an invariant measure \(\nu \) obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the restriction \(f|_\Lambda \). This allows then to obtain formulas for the measure \(\nu \) of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of \(\nu \).

Homoclinic classes with shadowing

Abstract We show that for C 1 generic diffeomorphisms, an isolated homoclinic class is shadow-able if and only if it is a hyperbolic basic set. Mathematics Subject Classification 2000: 37C20; 37C05; 37C29; 37D05.

Diffeomorphisms with various C1 stable properties

Let M be a smooth compact manifold and Λ be a compact invariant set. In this article, we prove that, for every robustly transitive set Λ, f|∧ satisfies a C1-genericstable shadowable property (resp., C1-generic-stable transitive specification property or C1-generic-stable barycenter property) if and only if Λ is a hyperbolic basic set. In particular, f|∧ satisfies a C1-stable shadowable property (resp., C1-stable transitive specification property or C1-stable barycenter property) if and only if Λ is a hyperbolic basic set. Similar results are valid for volume-preserving case.

Nonuniform hyperbolicity for C 1-generic diffeomorphisms

We study the ergodic theory of non-conservative C 1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C 1-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C 1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ. In addition, confirming a claim made by R. Mañé in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin’s Stable Manifold Theorem, even if the diffeomorphism is only C 1.

Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups

Assume that $X$ is a hyperbolic basic set for $f:X\to X$.We show new examples of Lie group fibers $G$ for which, in the class of $C^r, r>0,$ $G$-extensionsof $f$, those that are transitive are open and dense. The fibers are semidirectproducts of compact and nilpotent groups.

Geodesic flows with positive topological entropy, twist maps and hyperbolicity

We prove a perturbation lemma for the derivative of geodesic flows in high dimension. This implies that a C 2 generic riemannian metric has a nontrivial hyperbolic basic set in its geodesic flow.

Arc exchange systems and renormalization

We exhibit the construction of stable arc exchange systems from the stable laminations of hyperbolic diffeomorphisms. We prove a one-to-one correspondence between (i) Lipshitz conjugacy classes of C 1+H stable arc exchange systems that are C 1+H fixed points of renormalization and (ii) Lipshitz conjugacy classes of C 1+H diffeomorphisms f with hyperbolic basic sets Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. Let HD s (Λ) and HD u (Λ) be, respectively, the Hausdorff dimension of the stable and unstable leaves intersected with the hyperbolic basic set Λ. If HD u (Λ) = 1, then the Lipschitz conjugacy is, in fact, a C 1+H conjugacy in (i) and (ii). We prove that if the stable arc exchange system is a fixed point of renormalization with bounded geometry, then the stable arc exchange system is smooth conjugate to an affine stable arc exchange system.

Transitivity of Euclidean-type extensions of hyperbolic systems

AbstractLetf:X→Xbe the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class ofCr(r>0) cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for odd values ofn≥3. More generally, we consider Euclidean-type groupsG⋉ℝnwhereGis a compact connected Lie group acting linearly on ℝn. When Fix G={0}, it is again the case that the transitive cocycles are residual. When Fix G≠{0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.

Stability of mixing and rapid mixing for hyperbolic flows

We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic flows. Amongst C r Axiom A flows, r ≥ 2, we show that there is a C 2 -open, C r -dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing. This is the first general result on the stability of rapid mixing (or even mixing) for Axiom A � � �