American Institute of Mathematical Sciences

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  2018, 25: 8-15. doi: 10.3934/era.2018.25.002

Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds

Mohammadreza Molaei

Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran

We express our thanks to anonymous referee for his/her valuable comments

Received  July 21, 2017 Published  April 2018

We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part of tangent space (at each point of this set) to two unstable and stable subspaces with exponentially increasing and exponentially decreasing dynamics on them. We prove the continuity of this decomposition via the metric created by a torsion-free pseudo-Riemannian connection. We present a global attractor for a diffeomorphism on an open submanifold of the hyperbolic space $H^2(1)$ which is not a hyperbolic set for it.

Keywords: hyperbolic set, pseudo-Riemannian manifold.
Mathematics Subject Classification: Primary: 37D05; Secondary: 53B30.
Citation: Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002
References:
[1]

V. M. Alekseev and M. Yakobson, Symbolic dynamics and hyperbolic dynamical systems, Phys. Rep., 75 (1981), 287-325.  doi: 10.1016/0370-1573(81)90186-1.  Google Scholar

[2]

V. Araujo and M. Viana, Hyperbolic dynamical systems, in Mathematics of Complexity and Dynamical Systems, Vols. 13, Springer, New York, 2012, 740-754.  Google Scholar

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C. Bona and J. Massó, Hyperbolic evolution system for numerical relativity, Phys. Rev. Lett., 68 (1992), 1097-1099.  doi: 10.1103/PhysRevLett.68.1097.  Google Scholar

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Y. Choquet-Bruhat and T. Ruggeri, Hyperbolicity of the 3+1 system of Einstein equations, Commun. Math. Phys., 89 (1983), 269-275.  doi: 10.1007/BF01211832.  Google Scholar

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S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973.  Google Scholar

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A. Mukherjee, Differential Topology, Springer International Publishing AG Switzerland, 2015 doi: 10.1007/978-3-319-19045-7.  Google Scholar

[10] J. Palis Jr and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York-Berlin, 1982.   Google Scholar
[11]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors, Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993.  Google Scholar

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H. Poincaré, Sur le probléme des trois corps et les equations de la dynamique, Acta Mathematica, 13 (1890), 1-270.   Google Scholar

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C. Ragazzo and L. S. Ruiz, Dynamics of an isolated, viscoelastic, self-gravitating body, Celestial Mech. Dynam. Astronom., 122 (2015), 303-332.  doi: 10.1007/s10569-015-9620-9.  Google Scholar

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

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R. Yang and J. Qi, Dynamics of generalized tachyon field, Eur. Phys. J. C, 72 (2012), 2095.  doi: 10.1140/epjc/s10052-012-2095-x.  Google Scholar

show all references

References:
[1]

V. M. Alekseev and M. Yakobson, Symbolic dynamics and hyperbolic dynamical systems, Phys. Rep., 75 (1981), 287-325.  doi: 10.1016/0370-1573(81)90186-1.  Google Scholar

[2]

V. Araujo and M. Viana, Hyperbolic dynamical systems, in Mathematics of Complexity and Dynamical Systems, Vols. 13, Springer, New York, 2012, 740-754.  Google Scholar

[3]

C. Bona and J. Massó, Hyperbolic evolution system for numerical relativity, Phys. Rev. Lett., 68 (1992), 1097-1099.  doi: 10.1103/PhysRevLett.68.1097.  Google Scholar

[4]

Y. Choquet-Bruhat and T. Ruggeri, Hyperbolicity of the 3+1 system of Einstein equations, Commun. Math. Phys., 89 (1983), 269-275.  doi: 10.1007/BF01211832.  Google Scholar

[5]

A. Gogolev, Bootstrap for local rigidity of Anosov automorphisms on the 3-torus, Commun. Math. Phys., 352 (2017), 439-455.  doi: 10.1007/s00220-017-2863-4.  Google Scholar

[6]

J. S. Hadamard, Sur l'it$\acute{e}ration$ et les solutions asymptotiques des équations différentielles, Bulletin de la Société Mathématique de France, 29 (1901), 224-228.   Google Scholar

[7]

B. Hasselblatt, Hyperbolic dynamical systems, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 239-319. doi: 10.1016/S1874-575X(02)80005-4.  Google Scholar

[8]

S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973.  Google Scholar

[9]

A. Mukherjee, Differential Topology, Springer International Publishing AG Switzerland, 2015 doi: 10.1007/978-3-319-19045-7.  Google Scholar

[10] J. Palis Jr and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York-Berlin, 1982.   Google Scholar
[11]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors, Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993.  Google Scholar

[12]

H. Poincaré, Sur le probléme des trois corps et les equations de la dynamique, Acta Mathematica, 13 (1890), 1-270.   Google Scholar

[13]

C. Ragazzo and L. S. Ruiz, Dynamics of an isolated, viscoelastic, self-gravitating body, Celestial Mech. Dynam. Astronom., 122 (2015), 303-332.  doi: 10.1007/s10569-015-9620-9.  Google Scholar

[14]

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

[15]

R. Yang and J. Qi, Dynamics of generalized tachyon field, Eur. Phys. J. C, 72 (2012), 2095.  doi: 10.1140/epjc/s10052-012-2095-x.  Google Scholar

Figure 1.  The hyperbolic space $H^2(1)$
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Figure 2.  The black circle $C$ is a global attractor for $h$ but it is not a hyperbolic set for it
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