[Paper Review] On the abundance of SRB measures

Yongluo Cao, Zeya Mi+1 more|arXiv (Cornell University)|Apr 10, 2018

Mathematical Dynamics and Fractals7 references3 citations

TL;DR

This paper establishes the abundance of Sinai-Ruelle-Bowen (SRB) measures for $C^1$ diffeomorphisms away from those with homoclinic tangencies, using random perturbations and layered analysis of Gibbs $cu$-states. The key result shows that in $\mathrm{Diff}^1(M)$, any diffeomorphism can be approximated by ones with SRB measures, homoclinic tangencies, or essentially Morse-Smale systems, supporting Palis' conjecture on physical measures in global dynamics.

ABSTRACT

We prove the abundance of Sinai-Ruelle-Bowen measures for diffeomorphisms away from ones with a homoclinic tangency. This is motivated by conjectures of Palis on the existence of physical (Sinai-Ruelle-Bowen) measures for global dynamics. The main novelty in this paper is that we have to deeply study Gibbs $cu$-states in different levels. Note that we have to use random perturbations to give some upper bound of the level of Gibbs $cu$-states.

Motivation & Objective

• To support Palis' conjecture that most dissipative diffeomorphisms have finitely many physical (SRB) measures whose basins cover full Lebesgue measure.
• To establish the abundance of SRB measures in the $C^1$ topology for diffeomorphisms not exhibiting homoclinic tangencies.
• To analyze the structure of Gibbs $cu$-states across multiple levels of domination in partially hyperbolic dynamics.
• To prove the existence of SRB measures on attracting sets with one-dimensional dominated center bundles in $C^2$ diffeomorphisms.

Proposed method

• Uses random perturbations to control and bound the level of Gibbs $cu$-states, enabling upper estimates in the analysis of invariant measures.
• Applies a layered approach to study Gibbs $cu$-states by analyzing increasingly refined invariant bundles in dominated splittings.
• Employs extended dynamical systems and lifted measures to study stationary measures and their disintegration along unstable manifolds.
• Utilizes a Pliss-like lemma to extract sets of positive density with uniform lower bounds on Lyapunov exponents.
• Applies absolute continuity of foliations and conditional measures to relate Lebesgue and Gibbs measures along unstable manifolds.
• Relies on the theory of Pesin blocks and measurable partitions subordinate to unstable manifolds to construct physical measures.

Experimental results

Research questions

• RQ1 Can SRB measures be shown to be dense in the space of $C^1$ diffeomorphisms away from homoclinic tangencies?
• RQ2 How can Gibbs $cu$-states be analyzed across multiple levels of domination to ensure the existence of SRB measures?
• RQ3 To what extent do random perturbations help in controlling the statistical properties of invariant measures in non-uniformly hyperbolic systems?
• RQ4 Under what conditions does a partially hyperbolic attracting set with one-dimensional center bundles support an SRB measure?
• RQ5 Is the set of diffeomorphisms with SRB measures dense in $\mathrm{Diff}^1(M)$, as predicted by Palis?

Key findings

• In $\mathrm{Diff}^1(M)$, any diffeomorphism can be $C^1$-approximated by one with a homoclinic tangency, an essentially Morse-Smale system, or a system with an SRB measure.
• The set of $C^1$ diffeomorphisms with measures satisfying the Pesin entropy formula is dense in $\mathrm{Diff}^1(M)$, excluding only those with homoclinic tangencies.
• For $C^2$ diffeomorphisms with a partially hyperbolic attracting set and one-dimensional center bundles, SRB measures exist under the given dynamical conditions.
• The disintegration of Gibbs $cu$-states along unstable manifolds is absolutely continuous with respect to Lebesgue measure, enabling the construction of physical measures.
• The use of random perturbations allows for upper bounds on the level of Gibbs $cu$-states, crucial for controlling the statistical behavior of the system.
• A Pliss-like lemma is established to extract sets of density $1 - \varepsilon$ where Lyapunov exponents are uniformly bounded below, supporting the existence of positive metric entropy measures.