SRB measures for pointwise hyperbolic systems on open regions
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. ARTICLES .
September 2020 Vol. 63 No. 9: 1671–1720 https://doi.org/10.1007/s11425-020-1766-3
SRB measures for pointwise hyperbolic systems on open regions Dedicated to the Memory of Professor Shantao Liao
Jianyu Chen1 , Huyi Hu2,∗ & Yunhua Zhou3 1School
of Mathematical Sciences & Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, China; 2Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA; 3College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China Email: [email protected], [email protected], [email protected] Received July 24, 2020; accepted August 19, 2020; published online August 25, 2020
Abstract
A diffeomorphism f : M → M is pointwise partially hyperbolic on an open invariant subset N ⊂ M
if there is an invariant decomposition TN M = E u ⊕ E c ⊕ E s such that Dx f is strictly expanding on Exu and contracting on Exs at each x ∈ N . We show that under certain conditions f has unstable and stable manifolds, and admits a finite or an infinite u-Gibbs measure µ. If f is pointwise hyperbolic on N , then µ is a SinaiRuelle-Bowen (SRB) measure or an infinite SRB measure. As applications, we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok’s map have the properties. Keywords
pointwise hyperbolicity, unstable manifold, SRB measure, almost Anosov diffeomorphism, Katok’s
map MSC(2010)
37C40, 37D30, 37D50
Citation: Chen J Y, Hu H Y, Zhou Y H. SRB measures for pointwise hyperbolic systems on open regions. Sci China Math, 2020, 63: 1671–1720, https://doi.org/10.1007/s11425-020-1766-3
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Introduction
Let M be a C ∞ compact Riemannian manifold with dimension d > 2 and f : M → M be a C r , r > 1, diffeomorphism. Let N be a connected open f -invariant subset of M . We say that f is pointwise hyperbolic if there is a decomposition of the tangent bundle TN M into a direct sum of a stable bundle E s and a unstable bundle E u that are invariant under Df such that restricted to E u and E s , Df is expanding and contracting respectively. Pointwise partial hyperbolicity can be understood in a similar way (see Definition 2.1 for precise definition). Since N is open, typically hyperbolicity becomes weak near the boundary of N . In this case the hyperbolicity cannot be uniform. The formal definition of pointwise partial hyperbolic diffeomorphisms was first introduced in a work by Burns and Wilkinson [7] concerning ergodicity of volume preserving partial hyperbolic diffeomorphisms. Since the diffeomorphisms are defined on compact manifolds, the expansion and contraction are actually uniform. Katok’s map [19] on M = D2 is one of early examples of pointwise hyperbolic diffeomorphisms in * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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which the expansion and contraction can be arbitrarily slow near N = M \{p, ∂D2 }, where p is a fixed point of
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