Some geometrical models of chaotic dynamics

The free motion of a particle on a surface of constant negative curvature (a pseudosphere) was one of the first models of chaotic motion. It became the prototype for the theory of hyperbolic systems developed by Bowen and Sinai. In these models, geometry suggests a symbolic coding which already exhibits fully chaotic behaviour. One can return to these models to seek possible manifestations of quantum chaos. Here the mathematical technique is harmonic analysis on hyperbolic space. Chaotic behaviour seems to appear both in the behaviour of individual eigenfunctions and in the sequence of spectral values.

[1]  G. A. Hedlund.  A Metrically Transitive Group Defined by the Modular Groups , 1935 .

[2]  Dennis Sullivan,et al.  On The Ergodic Theory at Infinity of an Arbitrary Discrete Group of Hyperbolic Motions , 1981 .

[3]  C. Series.  Geometrical Markov coding of geodesics on surfaces of constant negative curvature , 1986 .

[4]  N. Balazs,et al.  Spectral fluctuations and zeta functions , 1987 .

[5]  S M Voronin,et al.  THEOREM ON THE “UNIVERSALITY” OF THE RIEMANN ZETA-FUNCTION , 1975 .

[6]  D. Hejhal.  The Selberg trace formula for PSL (2, IR) , 1983 .

[7]  M. Gutzwiller.  Stochastic behavior in quantum scattering , 1983 .

[8]  A. Reich.  Werteverteilung von Zetafunktionen , 1980 .

[9]  S. Helgason.  Groups and geometric analysis , 1984 .

[10]  G. Iooss,et al.  Chaotic behaviour of deterministic systems , 1983 .

[11]  Y. Sinai,et al.  Markov partitions and C-diffeomorphisms , 2020, Hamiltonian Dynamical Systems.

[12]  G. A. Hedlund.  On the Metrical Transitivity of the Geodesics on Closed Surfaces of Constant Negative Curvature , 1934 .

[13]  Harold Marston Morse.  A One-to-One Representation of Geodesics on a Surface of Negative Curvature , 1921 .

[14]  N. Balazs,et al.  Chaos on the pseudosphere , 1986 .

[15]  Rufus Bowen,et al.  SYMBOLIC DYNAMICS FOR HYPERBOLIC FLOWS. , 1973 .

[16]  E. Artin.  Ein mechanisches system mit quasiergodischen bahnen , 1924 .

[17]  M. Ratner.  Markov partitions for anosov flows onn-dimensional manifolds , 1973 .

[18]  D. Ruelle,et al.  Resonances of chaotic dynamical systems. , 1986, Physical review letters.

[19]  D. Ruelle.  Locating resonances for AxiomA dynamical systems , 1986 .