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

Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds

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.

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.

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

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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.

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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.

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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.

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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. 

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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.

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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.

[9]

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

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

[12]

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

[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.

[14]

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

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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.

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.

[2]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[9]

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

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

[12]

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

[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.

[14]

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

[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.

Figure 1.  The hyperbolic space $H^2(1)$
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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