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On the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with compact center leaves

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Joas Elias Rocha
Joas Elias Rocha
Joas Elias Rocha
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Ali Tahzibi at University of São Paulo
Ali Tahzibi
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Abstract

In this paper we study the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms defined on 3-torus with compact center leaves. Assuming the existence of a periodic leaf with Morse–Smale dynamics we prove a sharp upper bound for the number of maximal measures in terms of the number of sources and sinks of Morse–Smale dynamics. A well-known class of examples for which our results apply are the so called Kan-type diffeomorphisms admitting physical measures with intermingled basins.
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Mathematische Zeitschrift (2022) 301:471–484
https://doi.org/10.1007/s00209-021-02925-1
Mathematische Zeitschrift
On the number of ergodic measures of maximal entropy for
partially hyperbolic diffeomorphisms with compact center
leaves
Joas Elias Rocha2
·Ali Tahzibi1
Received: 14 June 2021 / Accepted: 18 October 2021 / Published online: 15 January 2022
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
Abstract
In this paper we study the number of ergodic measures of maximal entropy for partially
hyperbolic diffeomorphisms defined on 3-torus with compact center leaves. Assuming the
existence of a periodic leaf with Morse–Smale dynamics we prove a sharp upper bound for
the number of maximal measures in terms of the number of sources and sinks of Morse–
Smale dynamics. A well-known class of examples for which our results apply are the so
called Kan-type diffeomorphisms admitting physical measures with intermingled basins.
1 Introduction
Measures of maximal entropy (M.M.E) are global maxima for the Kolmogorov–Sinai entropy
map μ→hμ(f)where f:M→Mis a continuous transformation defined on a compact
metric space M. In this paper by m.m.e we mean ergodic measures of maximal entropy. These
measures, when exist, are considered among natural invariant measures of f. Newhouse [17]
studied an upper bound on the defect in uppersemicontinuity of topological and metric entropy
and proved upper semi continuity of μ→hμ(f)under C∞regularity condition on f.As a
consequence any C∞diffeomorphism of a compact manifold admits at least one measure of
maximal entropy.
However, there are examples of (with high regularity Cr,0<r<∞) diffeomorphisms
without any measure of maximal entropy (See for instance [6,16].)
Ali Tahzibi is supported by FAPESP 107/06463-3 and CNPq (PQ) 303025/2015-8.
BAli Tahzibi
tahzibi@icmc.usp.br
Joas Elias Rocha
joas.rocha@ufrpe.br
1Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo (USP), São Carlos,
Brazil
2Unidade Acadêmica de Cabo de Santo Agostinho, Universidade Federal Rural de Pernambuco, Recife,
Brazil
123
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