Abstract
In this paper, we provide a technique result on the existence of Gibbs cu-states for diffeomorphisms with dominated splittings. More precisely, for given $C^2$ diffeomorphim $f$ with dominated splitting $T_{\Lambda}M = E\oplus F$ on an attractor $\Lambda$, by considering some suitable random perturbation of $f$, we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle $E$, then its ergodic components contain Gibbs cu-states associated to $E$. With this technique, we show the existence of SRB measures and physical measures for some systems exhibitting dominated splittings and weak hyperbolicity.
Highlights
Introduction and main resultsOur aim of this paper is to show the existence of SRB measures for some systems beyond hyperbolicity
SRB measures were discovered by Sinai, Ruelle and Bowen in 1970s for uniformly hyperbolic systems [27, 8, 7, 25]
The SRB theory rapidly became a central topic of dynamical systems
Summary
Our aim of this paper is to show the existence of SRB (or physical) measures for some systems beyond hyperbolicity. Note that without using the tool of random perturbations, [11] got the existence of invariant measures satisfying Pesin entropy formula for more general dynamical systems (see [35] ). In contrast to their strategy, in the proof of Theorem B or Theorem A, we do not involve the discussion on Pesin entropy formula, and different to them, we consider special zero noise limits—randomly ergodic limits, and take advantage of this modification, as ergodic stationary measures inherit more information from their limit than stationary measures. The proofs of Theorem A and Theorem B are provided in §6, in which some applications of them are studied
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