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Measures of intermediate pressures for geometric Lorenz attractors

Yi Shi, Xiaodong Wang

Abstract

Pressure measures the complexity of a dynamical system concerning a continuous observation function. A dynamical system is called to admit the intermediate pressure property if for any observation function, the measure theoretical pressures of all ergodic measures form an interval. We prove that the intermediate pressure property holds for $C^r (r\geq 2)$ generic geometric Lorenz attractors while it fails for $C^r (r\geq 2)$ dense geometric Lorenz attractors, which gives a sharp contrast. Similar results hold for $C^1$ singular hyperbolic attractors.

Measures of intermediate pressures for geometric Lorenz attractors

Abstract

Pressure measures the complexity of a dynamical system concerning a continuous observation function. A dynamical system is called to admit the intermediate pressure property if for any observation function, the measure theoretical pressures of all ergodic measures form an interval. We prove that the intermediate pressure property holds for generic geometric Lorenz attractors while it fails for dense geometric Lorenz attractors, which gives a sharp contrast. Similar results hold for singular hyperbolic attractors.

Paper Structure

This paper contains 15 sections, 11 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Topological pressure
  4. Geometric Lorenz attractor and singular hyperbolicity
  5. Ergodic theory for singular hyperbolic homoclinic classes
  6. Preliminary lemmas for entropy
  7. Ergodic measures with intermediate pressure
  8. Case I: $P< P_{\mu}(X,\varphi)$.
  9. Subcase I.1: $\int \varphi {\rm d}\mu<P< P_{\mu}(X,\varphi)$.
  10. Subcase I.2: $\int \varphi {\rm d}\mu=P< P_{\mu}(X,\varphi)$.
  11. Case II: $P= P_{\mu}(X,\varphi)$.
  12. Subcase II.1: $\int \varphi {\rm d}\mu<P= P_{\mu}(X,\varphi)$.
  13. Subcase II.2: $\int \varphi {\rm d}\mu=P= P_{\mu}(X,\varphi)$.
  14. Proof of Theorem \ref{['Thm:main-intermediate-pressure']}
  15. Dense part of Theorem \ref{['Thm:Lorenz']} & \ref{['Thm:SH-attractor']}

Key Result

Theorem A

For every $r\in \mathbb{N}_{\geq 2}\cup\{\infty\}$, there exist a dense $G_\delta$ subset $\mathcal{R}^r$ and a dense subset $\mathcal{D}^r$ in $\mathscr{X}^r(M^3)$ such that

Figures (1)

  • Figure 1: Geometric Lorenz attractor and the return map

Theorems & Definitions (31)

  • Conjecture 1.1: Katok
  • Theorem A
  • Remark 1.1
  • Theorem B
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 3.1
  • ...and 21 more