Oscillating Statistics of Transitive Dynamics
Author(s)
Eleonora Catsigeras
Affiliation(s)
Instituto de Matemática y Estadística “Rafael Laguardia”, Universidad de la República, Montevideo, Uruguay.
Instituto de Matemática y Estadística “Rafael Laguardia”, Universidad de la República, Montevideo, Uruguay.
ABSTRACT
We prove that topologically generic orbits of C0 , transitive and non-uniquely ergodic dynamical systems, exhibit an extremely oscillating asymptotical statistics. Precisely, the minimum weak* compact set of invariant probabilities that describes the asymptotical statistics of each orbit of a residual set contains all the ergodic probabilities. If besides f is ergodic with respect to the Lebesgue measure, then also Lebesgue-almost all the orbits exhibit that kind of extremely oscillating statistics.
We prove that topologically generic orbits of C0 , transitive and non-uniquely ergodic dynamical systems, exhibit an extremely oscillating asymptotical statistics. Precisely, the minimum weak* compact set of invariant probabilities that describes the asymptotical statistics of each orbit of a residual set contains all the ergodic probabilities. If besides f is ergodic with respect to the Lebesgue measure, then also Lebesgue-almost all the orbits exhibit that kind of extremely oscillating statistics.
Cite this paper
Catsigeras, E. (2015) Oscillating Statistics of Transitive Dynamics. Advances in Pure Mathematics, 5, 534-543. doi: 10.4236/apm.2015.59049.
Catsigeras, E. (2015) Oscillating Statistics of Transitive Dynamics. Advances in Pure Mathematics, 5, 534-543. doi: 10.4236/apm.2015.59049.
References
[1] Abdenur, F. and Andersson, M. (2013) Ergodic Theory of Generic Continuous Maps. Communications in Mathematical Physics, 318, 831-855. [2] Bonatti, C. (2011) Towards a Global View of Dynamical Systems, for the C1-Topology. Ergod. Ergodic Theory and Dynamical Systems, 31, 959-993. http://dx.doi.org/10.1017/S0143385710000891 [3] Buzzi, J. (2011) Chaos and Ergodic Theory. In: Meyers, R.A., Ed., Mathematics of Complexity and Dynamical Systems, Springer, New York, 63-87. [4] Catsigeras, E. and Enrich, H. (2011) SRB-Like Measures for C0 Dynamics. Bulletin of the Polish Academy of Sciences Mathematics, 59, 151-164. http://dx.doi.org/10.4064/ba59-2-5 [5] Liverani, C. (2013) Multidimensional Expanding Maps with Singularities: A Pedestrian Approach. Ergodic Theory and Dynamical Systems, 33, 168-182. http://dx.doi.org/10.1017/S0143385711000939 [6] Mañé, R. (1987) Ergodic Theory and Differentiable Dynamics. Springer-Verlag, New York. [7] Walters, P. (1982) An Introduction to Ergodic Theory. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-5775-2 [8] Viana, M. and Yang, J. (2013) Physical Measures and Absolute Continuity for One-Dimensional Center Direction. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 30, 845-877.
[1] Abdenur, F. and Andersson, M. (2013) Ergodic Theory of Generic Continuous Maps. Communications in Mathematical Physics, 318, 831-855. [2] Bonatti, C. (2011) Towards a Global View of Dynamical Systems, for the C1-Topology. Ergod. Ergodic Theory and Dynamical Systems, 31, 959-993. http://dx.doi.org/10.1017/S0143385710000891 [3] Buzzi, J. (2011) Chaos and Ergodic Theory. In: Meyers, R.A., Ed., Mathematics of Complexity and Dynamical Systems, Springer, New York, 63-87. [4] Catsigeras, E. and Enrich, H. (2011) SRB-Like Measures for C0 Dynamics. Bulletin of the Polish Academy of Sciences Mathematics, 59, 151-164. http://dx.doi.org/10.4064/ba59-2-5 [5] Liverani, C. (2013) Multidimensional Expanding Maps with Singularities: A Pedestrian Approach. Ergodic Theory and Dynamical Systems, 33, 168-182. http://dx.doi.org/10.1017/S0143385711000939 [6] Mañé, R. (1987) Ergodic Theory and Differentiable Dynamics. Springer-Verlag, New York. [7] Walters, P. (1982) An Introduction to Ergodic Theory. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-5775-2 [8] Viana, M. and Yang, J. (2013) Physical Measures and Absolute Continuity for One-Dimensional Center Direction. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 30, 845-877.