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Stable ergodicity for partially hyperbolic attractors with negative central exponents. (English) Zbl 1145.37021

This work is devoted to the stable ergodicity with respect to the “natural” measure supported on a partially hyperbolic attracting set. To be more precise, the authors consider a \(C^{1+\alpha}\) diffeomorphims \(f\) defined on a compact smooth Riemannian manifold \(M\), and they assume \(\Lambda\subseteq M\) be a partially hyperbolic attractor for \(f\). In this dissipative setting, the “natural” measures are the SRB-measures who play the role of Lebesgue measure in the conservative case.
It is well known that every partially hyperbolic attractor supports measures abosultely continuous along the unstable leaves (the authors call such measures as \(u\)-measures). That measures are SRB-measures if they are ergodic and they have negative Lyapunov exponents along the center direction. In this work, the authors provides a condition to guarantee that that SRB-measure is unique in a neigbourhood of \(f\).
This work is a continuation of [the first three authors, J. Stat. Phys. 108, No. 5–6, 927–942 (2002; Zbl 1124.37308)] where the authors deal with the conservative case.

MSC:

37D30 Partially hyperbolic systems and dominated splittings
37A25 Ergodicity, mixing, rates of mixing
37B25 Stability of topological dynamical systems

Citations:

Zbl 1124.37308
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