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

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

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  • 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.

    Mathematics Subject Classification: Primary: 37C40, 37D25, 37D30, 37H15.


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