R-robustly measure expansive homoclinic classes are hyperbolic


Manseob Lee - Department of Mathematics, Mokwon University, Daejeon, 302-729, Korea


Let \(f:M\to M\) be a diffeomorphism on a closed smooth \(n(n\geq 2)\)-dimensional manifold \(M\) and let \(p\) be a hyperbolic periodic point of \(f\). We show that if the homoclinic class \(H_f(p)\) is R-robustly measure expansive then it is hyperbolic.


Expansive, measure expansive, local product structure, shadowing, hyperbolic, homoclinic class, generic


[1] A. Arbieto, Periodic orbits and expansiveness, Math. Z., 269 (2011), 801–807.
[2] C. Bonatti, S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33–104.
[3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer-Verlag, Berlin, (2008).
[4] M. W. Hirsch, C. C. Pugh, M. Shub, Invariant manifolds, Lecture Note in Math., Springer-Verlag, New York, (1977).
[5] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576–598.
[6] N. Koo, K. Lee, M. Lee, Generic diffeomorphisms with measure expansive homoclinic classes, J. Difference Equ. Appl., 20 (2014), 228–236.
[7] M. Lee, Measure expansiveness of the generic view point, preprint.
[8] K. Lee, M. Lee, Hyperbolicity of \(C^1\)-stably expansive homoclinic classes, Discrete Contin. Dyn. Syst., 27 (2010), 1133– 1145.
[9] K. Lee, M. Lee, Measure expansive homoclinic classes, Osaka J. Math., 53 (2016), 873–887.
[10] X. Li, On R-robustly entropy-expansive diffeomorphisms, Bull. Braz. Math. Soc., 43 (2012), 73–98.
[11] R. Mañé, Expansive diffeomorphisms, Lecture Notes in Math., Springer, Berlin, (1975).
[12] R. Mañé, Contribution to stability conjecture, Topology, 17 (1978), 383–396.
[13] C. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst., 32 (2012), 293–301.
[14] C. A. Morales, V. F. Sirvent, Expansive measures, Instituto Nacional de Matemtica Pura e Aplicada (IMPA), Rio de Janeiro, (2013).
[15] M. J. Pacifico, E. R. Pujals, M. Sambarino, J. L. Vieites, Robustly expansive codimension-one homoclinic classes are hyperbolic, Ergodic Theory Dynam. Systems, 29 (2009), 179–200.
[16] M. J. Pacifico, E. R. Pujals, J. L. Vieites, Robustly expansive homoclinic classes, Ergodic Theory Dynam. Systems, 25 (2005), 271–300.
[17] M. J. Pacificao, J. L. Vieites, On measure expansive diffeomorphisms, Proc. Amer. Math. Soc., 143 (2015), 811–819.
[18] K. Sakai, C1-stably shadowable chain components, Ergodic Theory Dynam. Systems, 28 (2008), 987–1029.
[19] K. Sakai, N. Sumi, K. Yamamoto, Measure expansive diffeomorphisms, J. Math. Anal. Appl., 414 (2014), 546–552.
[20] M. Sambarino, J. Vieitez, On \(C^1\)-persistently expansive homoclinic classes, Discrete Contin. Dynam. Syst., 14 (2006), 465–481.
[21] M. Sambarino, J. Vieitez, Robustly expansive homoclinic classes are generically hyperbolic, Discrete Contin. Dynam. Syst., 24 (2009), 1325–1333.
[22] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769–774.
[23] X. Wen, S. Gan, L. Wen, \(C^1\)-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340–357.


XML export