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Ergodic properties of invariant measures for C1+α non-uniformly hyperbolic systems

Published online by Cambridge University Press:  08 February 2012

CHAO LIANG ,
WENXIANG SUN  and
XUETING TIAN
CHAO LIANG
Affiliation:
Applied Mathematical Department, The Central University of Finance and Economics, Beijing 100081, China (email: chaol@cufe.edu.cn)
WENXIANG SUN
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China (email: sunwx@math.pku.edu.cn)
XUETING TIAN
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Mathematical Sciences, Peking University, Beijing 100871, China (email: tianxt@amss.ac.cn, txt@pku.edu.cn)
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Abstract

For every ergodic hyperbolic measure ω of a C1+α diffeomorphism, there is an ω-full-measure set $\tilde {\Lambda }$ (the union of $\tilde \Lambda _l=\mathrm {supp}( \omega |_{\Lambda _{l}})$, the support sets of ω on each Pesin block Λl, l=1,2,…) such that every non-empty, compact and connected subset $V\subseteq \mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ coincides with Vf(x), where $\mathcal {M}_{\mathrm {inv}}(\tilde {\Lambda })$ denotes the space of invariant measures supported on $\tilde {\Lambda }$ and Vf(x) denotes the accumulation set of time averages of Dirac measures supported at one orbit of some point x. For each fixed set V, the points with the above property are dense in the support supp (ω) . In particular, points satisfying $V_f(x)=\mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ are dense in supp (ω) . Moreover, if supp (ω) is isolated, the points satisfying $V_f(x)\supseteq \mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ form a residual subset of supp (ω) . These extend results of K. Sigmund [On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285–299] (see also M. Denker, C. Grillenberger and K. Sigmund [Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, Berlin, Ch. 21]) from the uniformly hyperbolic case to the non-uniformly hyperbolic case. As a corollary, irregular + points form a residual set of supp (ω) .

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 560 - 584
Copyright
©2012 Cambridge University Press

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