[{"command":"insert","method":null,"selector":".course-modal-ajax-placeholder-957864","data":"\u003Cdiv class=\u0022container content-rte\u0022\u003E\n  \u003Cdiv class=\u0022row\u0022\u003E\n    \u003Cdiv class=\u0022box col-md-6 bg-primary text-white\u0022\u003E\n      \u003Ch2 class=\u0022h1 text-left\u0022\u003E  Th\u00e9orie ergodique 2025-2026\n\u003C\/h2\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EECTS :\n          7.5\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ESemestre calendaire :\n          Semester 1\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ED\u00e9tail du volume horaire\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ECours magistraux :\u003C\/span\u003E\n                                      \u003Cspan\u003E25\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                              \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ETravaux dirig\u00e9s :\u003C\/span\u003E\n                                      \u003Cspan\u003E12.5\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                                            \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ELangue d\u0026#039;enseignement\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EAnglais\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EEnseignement \u00e0 distance\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003Enon\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n    \u003Cdiv class=\u0022box col-md-6\u0022\u003E\n                    \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EProgramme \/ plan \/ contenus\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003EVersion fran\u00e7aise:\u003Cbr\u003E\nVocabulaire (transformations pr\u00e9servant une mesure, \u00e9quivalence, ensembles invariants, errance mesur\u00e9e, conservativit\u00e9) et exemples de base (rotations et d\u00e9calage de Bernoulli). Th\u00e9or\u00e8me de r\u00e9currence de Poincar\u00e9, th\u00e9or\u00e8me de Kac, temps de retour. Ergodicit\u00e9. Th\u00e9or\u00e8mes ergodiques en moyenne et ponctuel. Th\u00e9or\u00e8me ergodique sub-additif de Kingman et th\u00e9or\u00e8me ergodique multiplicatif d\u0027Oseledets. M\u00e9lange et m\u00e9lange exponentiel. Op\u00e9rateur de Koopman. Dynamique lin\u00e9aire mesur\u00e9e des tores. Entropie d\u0027une mesure invariante. Th\u00e9orie de l\u0027information de Shannon. D\u00e9calages de Markov.\u003C\/p\u003E\n\u003Cp\u003EEnglish version:\u003Cbr\u003E\nVocabulary (measure preserving transformations, invariant subsets, measured wandering, conservativity) and basic examples (rotations and Bernoulli shifts). Poincar\u00e9 recurrence theorem, return time, Kac theorem. Ergodicity. Mean and Birkhoff ergodic theorems. Kingman\u0027s subadditive ergodic theorem and Oseledets multiplicative ergodic theorem. Mixing and exponential mixing. Koopman operator. Measured linear dynamics of tori. Entropy of an invariant measure. Shannon\u0027s information theory. Markov shifts.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n                                        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EBibliographie\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003EF. Paulin, Introduction \u00e0 la th\u00e9orie ergodique et aux syst\u00e8mes dynamiques topologiques et diff\u00e9rentiables, Notes de cours Universit\u00e9 Paris-Saclay, https:\/\/www.imo.universite-paris-saclay.fr\/~frederic.paulin\/notescours\/liste_notescours.html\u003C\/p\u003E\n\u003Cp\u003EI. P. Cornfeld, S. V. Fomin et Ya. G. Sinai. Ergodic theory. Grund. Math. Wiss.\u003Cbr\u003E\n245, Springer-Verlag, 1982.\u003C\/p\u003E\n\u003Cp\u003EA. Katok et B. Hasselblatt, Introduction to the modern theory of\u003Cbr\u003E\ndynamical systems, Ency. Math. App. 54, Cambridge University Press, 1995.\u003C\/p\u003E\n\u003Cp\u003EK. Petersen : Ergodic Theory, Cambridge Univ. Press, 3rd ed. 1995.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ENature de l\u0026#039;\u00e9valuation\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EEvaluation Continue non Int\u00e9grale\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n  \u003C\/div\u003E\n\u003C\/div\u003E\n","settings":null}]