TY - JOUR

T1 - Dominated Pesin theory

T2 - convex sum of hyperbolic measures

AU - Bochi, Jairo

AU - Bonatti, Christian

AU - Gelfert, Katrin

N1 - Publisher Copyright:
© 2018, Hebrew University of Jerusalem.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure μ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class H(μ) of periodic orbits for the homoclinic relation, called its intersection class. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index, such as the dominated splitting, is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption.

AB - In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure μ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class H(μ) of periodic orbits for the homoclinic relation, called its intersection class. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index, such as the dominated splitting, is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption.

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U2 - 10.1007/s11856-018-1699-8

DO - 10.1007/s11856-018-1699-8

M3 - Article

AN - SCOPUS:85046808717

SN - 0021-2172

VL - 226

SP - 387

EP - 417

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -