Abstract
We show that $C^\infty $-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal entropy in general, and exactly one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale’s spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irreducible countable state Markov shifts.
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journal = {Ergodic Theory Dynam. Systems},
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volume = {34},
year = {2014},
number = {6},
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author = {Buzzi, J{é}r{ô}me},
title = {The almost {B}orel structure of diffeomorphisms with some hyperbolicity},
booktitle = {Hyperbolic Dynamics, Fluctuations and Large Deviations},
series = {Proc. Sympos. Pure Math.},
volume = {89},
pages = {9--44},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2015},
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mrreviewer = {Anthony Quas},
doi = {10.1090/pspum/089/01491},
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note = {in preparation},
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