# American Institute of Mathematical Sciences

November  2019, 39(11): 6441-6465. doi: 10.3934/dcds.2019279

## SRB measures for some diffeomorphisms with dominated splittings as zero noise limits

 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received  December 2018 Revised  April 2019 Published  August 2019

Fund Project: Zeya Mi was partially supported by NSFC 11801278 and The Startup Foundation for Introducing Talent of NUIST(Grant No. 2017r070).

In this paper, we provide a technical 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 exhibiting dominated splittings and weak hyperbolicity.

Citation: Zeya Mi. SRB measures for some diffeomorphisms with dominated splittings as zero noise limits. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6441-6465. doi: 10.3934/dcds.2019279
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