September  2017, 37(9): 4923-4941. doi: 10.3934/dcds.2017166

Hyperbolic sets that are not contained in a locally maximal one

Iguá 4225 Esq. Mataojo, C.P. 11400 Montevideo, Uruguay

Received  February 2014 Revised  February 2017 Published  June 2017

Fund Project: The autor was partially supported by CSIC

In this paper we study two properties related to the structure of hyperbolic sets. First we construct new examples answering in the negative the following question posed by Katok and Hasselblatt in [[12], p. 272]

Question. Let $\Lambda$ be a hyperbolic set, and let $V$ be an open neighborhood of $\Lambda$. Does there exist a locally maximal hyperbolic set $\widetilde{\Lambda}$ such that $\Lambda \subset \widetilde{\Lambda} \subset V $?

We show that such examples are present in linear Anosov diffeomorophisms of $\mathbb{T}^3$, and are therefore robust.

Also we construct new examples of sets that are not contained in any locally maximal hyperbolic set. The examples known until now were constructed by Crovisier in [7] and by Fisher in [9], and these were either in dimension equal or bigger than 4 or they were not transitive. We give a transitive and robust example in $\mathbb{T}^3$. And show that such examples cannot be build in dimension 2.

Citation: Adriana da Luz. Hyperbolic sets that are not contained in a locally maximal one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4923-4941. doi: 10.3934/dcds.2017166
References:
[1]

D. V. Anosov, On certain hyperbolic sets, Mat. Zametki, 87 (2010), 608-622. doi: 10.1134/S0001434610050020. Google Scholar

[2]

D. V. Anosov, Extension of zero-dimensional hyperbolic sets to locally maximal ones, Mat. Sb., 201 (2010), 935-946. doi: 10.1070/SM2010v201n07ABEH004097. Google Scholar

[3]

D. V. Anosov, Geodesic Flows on Riemannian Manifolds with Negative Curvature In Proc. Steklov Inst. Math. , volume 90. American Mathematical Society, Providence, RI, 1967. Google Scholar

[4]

Y. BenoistP. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, J. Amer. Math. Soc., 5 (1992), 33-74. doi: 10.2307/2152750. Google Scholar

[5]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, volume 102 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2005. certain Partially Hyperbolic, Derived from Anosov systems, to appear in Ergodic Theory Dynam. Systems. Google Scholar

[6]

K. Burns and K. Gelfert, Thermodynamics for geodesic flows of rank 1 surfaces, Discrete and Continuous Dynamics, 34 (2014), 1841-1872. doi: 10.3934/dcds.2014.34.1841. Google Scholar

[7]

S. Crovisier, Une remarque sur les ensembles hyperboliques localement maximaux, C. R. Math. Acad. Sci. Paris, 334 (2002), 401-404. doi: 10.1016/S1631-073X(02)02274-4. Google Scholar

[8]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., Special Issue: In honor of Grigory Margulis. Part 1, 3 (2007), 1037–1055, doi: 10.4310/PAMQ.2007.v3.n4.a9. Google Scholar

[9]

T. Fisher, Hyperbolic sets that are not locally maximal, Ergodic Theory and Dynamical Systems, 26 (2006), 1491-1509. doi: 10.1017/S0143385706000411. Google Scholar

[10]

T. Fisher, Transitive hyperbolic sets on surfaces, http://uamte.math.byu.edu/~tfisher/documents/papers/transsurface102209.pdf.
https://scholar.google.com/citations?viewop=view_citation&hl=en&user=RWO1s7IAAAAJ&citation_for_view=RWO1s7IAAAAJ:UebtZRa9Y70C.Google Scholar

[11]

S. G. Hancock, Construction of invariant sets for anosov diffeomorphisms, J. London Math. Soc., 18 (1978), 339-348. doi: 10.1112/jlms/s2-18.2.339. Google Scholar

[12]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187. Google Scholar

[13]

M. Hirsch, On invariant subsets of hyperbolic sets, Manuscripta Math., 68 (1990), 271-293. Google Scholar

[14]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. Google Scholar

[15]

R. Mañe, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8. Google Scholar

[16]

R. Mañe, Invariant sets of Anosovs diffeomorphisms, Inventiones mathematicae, 46 (1978), 147-152. doi: 10.1007/BF01393252. Google Scholar

[17]

R. Mañe, A proof of the $C^1$ stability conjecture, Publ. Math.I.H.E.S., 66 (1988), 161-210. Google Scholar

[18]

J. Palis, On the $Ω$-stability conjecture, Publ. Math. I.H.E.S., 66 (1988), 211-215. Google Scholar

[19]

A. Passeggi and M. Sambarino, Examples of minimal diffeomorphisms on the two torus semiconjugated to an ergodic translation, Fund. Math., 222 (2013), 63-97. doi: 10.4064/fm222-1-4. Google Scholar

[20]

M. Sambarino, Hiperbolicidad y Estabilidad Ediciones IVIC, Venezuela, ISBN 978-980-261-108-9.Google Scholar

[21]

J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in Contributions to Nonlinear Functional Analysis (eds. E. H. Zarantonello and Author 2), Academic Press, (1971), 565–601. Google Scholar

[22]

M. Shub, Topologically Transitive Diffeomorphisms on $\mathbb{T}^4$ Lecture Notes in Math. , Vol. 206, Springer-Verlag, 1971.Google Scholar

[23]

S. Smale, Diferentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

[24]

S. Smale, The $Ω$-stability theorem, Proc. A.M. S. Symp Pure Math, 14 (1970), 289-297. Google Scholar

show all references

References:
[1]

D. V. Anosov, On certain hyperbolic sets, Mat. Zametki, 87 (2010), 608-622. doi: 10.1134/S0001434610050020. Google Scholar

[2]

D. V. Anosov, Extension of zero-dimensional hyperbolic sets to locally maximal ones, Mat. Sb., 201 (2010), 935-946. doi: 10.1070/SM2010v201n07ABEH004097. Google Scholar

[3]

D. V. Anosov, Geodesic Flows on Riemannian Manifolds with Negative Curvature In Proc. Steklov Inst. Math. , volume 90. American Mathematical Society, Providence, RI, 1967. Google Scholar

[4]

Y. BenoistP. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, J. Amer. Math. Soc., 5 (1992), 33-74. doi: 10.2307/2152750. Google Scholar

[5]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, volume 102 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2005. certain Partially Hyperbolic, Derived from Anosov systems, to appear in Ergodic Theory Dynam. Systems. Google Scholar

[6]

K. Burns and K. Gelfert, Thermodynamics for geodesic flows of rank 1 surfaces, Discrete and Continuous Dynamics, 34 (2014), 1841-1872. doi: 10.3934/dcds.2014.34.1841. Google Scholar

[7]

S. Crovisier, Une remarque sur les ensembles hyperboliques localement maximaux, C. R. Math. Acad. Sci. Paris, 334 (2002), 401-404. doi: 10.1016/S1631-073X(02)02274-4. Google Scholar

[8]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., Special Issue: In honor of Grigory Margulis. Part 1, 3 (2007), 1037–1055, doi: 10.4310/PAMQ.2007.v3.n4.a9. Google Scholar

[9]

T. Fisher, Hyperbolic sets that are not locally maximal, Ergodic Theory and Dynamical Systems, 26 (2006), 1491-1509. doi: 10.1017/S0143385706000411. Google Scholar

[10]

T. Fisher, Transitive hyperbolic sets on surfaces, http://uamte.math.byu.edu/~tfisher/documents/papers/transsurface102209.pdf.
https://scholar.google.com/citations?viewop=view_citation&hl=en&user=RWO1s7IAAAAJ&citation_for_view=RWO1s7IAAAAJ:UebtZRa9Y70C.Google Scholar

[11]

S. G. Hancock, Construction of invariant sets for anosov diffeomorphisms, J. London Math. Soc., 18 (1978), 339-348. doi: 10.1112/jlms/s2-18.2.339. Google Scholar

[12]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187. Google Scholar

[13]

M. Hirsch, On invariant subsets of hyperbolic sets, Manuscripta Math., 68 (1990), 271-293. Google Scholar

[14]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. Google Scholar

[15]

R. Mañe, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8. Google Scholar

[16]

R. Mañe, Invariant sets of Anosovs diffeomorphisms, Inventiones mathematicae, 46 (1978), 147-152. doi: 10.1007/BF01393252. Google Scholar

[17]

R. Mañe, A proof of the $C^1$ stability conjecture, Publ. Math.I.H.E.S., 66 (1988), 161-210. Google Scholar

[18]

J. Palis, On the $Ω$-stability conjecture, Publ. Math. I.H.E.S., 66 (1988), 211-215. Google Scholar

[19]

A. Passeggi and M. Sambarino, Examples of minimal diffeomorphisms on the two torus semiconjugated to an ergodic translation, Fund. Math., 222 (2013), 63-97. doi: 10.4064/fm222-1-4. Google Scholar

[20]

M. Sambarino, Hiperbolicidad y Estabilidad Ediciones IVIC, Venezuela, ISBN 978-980-261-108-9.Google Scholar

[21]

J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in Contributions to Nonlinear Functional Analysis (eds. E. H. Zarantonello and Author 2), Academic Press, (1971), 565–601. Google Scholar

[22]

M. Shub, Topologically Transitive Diffeomorphisms on $\mathbb{T}^4$ Lecture Notes in Math. , Vol. 206, Springer-Verlag, 1971.Google Scholar

[23]

S. Smale, Diferentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

[24]

S. Smale, The $Ω$-stability theorem, Proc. A.M. S. Symp Pure Math, 14 (1970), 289-297. Google Scholar

Figure 1.  A n-$ \varepsilon $-relation between $ x $ and $ y $
Figure 2.  Perturbing a neighborhood of $q$
Figure 3.  $H$ acting on the foliations
Figure 4.  Separation between the unstable leafs
Figure 5.  The tube $V$
Figure 6.  A n-$ \varepsilon $-relation between $ (0,0,0) $ and $ x_j $
Figure 7.  the local connected component of $x$ and $ p^s_{x} (lcc(x)) $
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