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Denseness of zero entropy aperiodic ergodic measures

Camila Crispin, Lorenzo J. Díaz

Abstract

We study partially hyperbolic homoclinic classes of $C^1$-generic diffeomorphisms with a one-dimensional central bundle, so that the central Lyapunov exponent $χ^c(μ)$ is well defined for any ergodic measure $μ$ supported on the class. We focus on nonhyperbolic homoclinic classes supporting ergodic measures with positive, zero, and negative central exponents. For each $α$ and a nontrivial homoclinic class $H$ of a $C^1$-generic diffeomorphism $f$, we consider the level set of measures \[ \mathcal{M}^α_{\mathrm{erg}}(f,H)= \left\{\text{$μ$ ergodic, supported on $H$, with } χ^c(μ)=α\right\}. \] In this generic setting, the range of $α$ for which $\mathcal{M}^α_{\mathrm{erg}}(f,H)$ is nonempty forms a nontrivial closed interval $I$. Since the set of periodic measures is countable, most of these sets contain no periodic measures. We show that for every $α$ in the interior of $I$, the so-called Axiomatized GIKN measures, a class of low-complexity, zero-entropy measures, are dense in $\mathcal{M}^α_{\mathrm{erg}}(f,H)$. This result can be viewed as an analogue of Sigmund's classical density of periodic measures for systems with the specification property, obtained here in a setting where the specification property does not hold and periodic measures are typically absent (in the considered level sets). We also present a similar result for the open class of blender-minimal diffeomorphisms, contained in the class of $C^1$-robustly transitive ones.

Denseness of zero entropy aperiodic ergodic measures

Abstract

We study partially hyperbolic homoclinic classes of -generic diffeomorphisms with a one-dimensional central bundle, so that the central Lyapunov exponent is well defined for any ergodic measure supported on the class. We focus on nonhyperbolic homoclinic classes supporting ergodic measures with positive, zero, and negative central exponents. For each and a nontrivial homoclinic class of a -generic diffeomorphism , we consider the level set of measures In this generic setting, the range of for which is nonempty forms a nontrivial closed interval . Since the set of periodic measures is countable, most of these sets contain no periodic measures. We show that for every in the interior of , the so-called Axiomatized GIKN measures, a class of low-complexity, zero-entropy measures, are dense in . This result can be viewed as an analogue of Sigmund's classical density of periodic measures for systems with the specification property, obtained here in a setting where the specification property does not hold and periodic measures are typically absent (in the considered level sets). We also present a similar result for the open class of blender-minimal diffeomorphisms, contained in the class of -robustly transitive ones.

Paper Structure

This paper contains 29 sections, 30 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Nonhyperbolic homoclinic classes
  3. Blender-minimal diffeomorphisms
  4. Framework, main definitions, and tools
  5. Axiomatized GIKN measures
  6. Homoclinic classes of $C^1$-generic diffeomorphisms
  7. Partial Hyperbolicity and strong foliations
  8. Ergodic measures and center Lyapunov exponents
  9. Blender-horseshoes
  10. Split flip-flop configurations
  11. Density of nonhyperbolic aperiodic GIKN measures
  12. Proof of Theorem \ref{['t.densityisolatedset']}
  13. Proof of Proposition \ref{['p.mainprop']}
  14. Proof of Lemma \ref{['l.periodicpointshomoclinicclass']}
  15. Liao-Gan's shadowing lemma
  16. ...and 14 more sections

Key Result

Theorem A

There exists a residual set $\mathcal{R}(M)$ of $\mathrm{Diff}^1(M)$ such that for every $f \in \mathcal{R}(M)$ and every homoclinic class $H(p_f)\in \mathrm{ISPH}(f)$, there exists a neighborhood $\mathcal{U}_f$ of $f$ such that for every $g \in \mathcal{R}(M) \cap \mathcal{U}_f$ and every $\alpha \in (\alpha_{\inf}(H(p_g)),\alpha_{\sup}(H(p_g)))$.

Theorems & Definitions (97)

  • Definition 1.2: Homoclinic classes in $\mathrm{ISPH}(f)$
  • Theorem A
  • Definition 1.3
  • Theorem B
  • Definition 2.1: $(\epsilon,\pi)$-good points
  • Definition 2.2: $(\epsilon,\rho)$-good approximations
  • Definition 2.3: GIKN sequences
  • Theorem 2.4
  • Definition 2.5: Axiomatized GIKN measures
  • Remark 2.6
  • ...and 87 more