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 and Continuous Dynamical Systems, 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

[23]

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

[24]

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

show all references

References:
[1]

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

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

[23]

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

[24]

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

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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