Abstract
This chapter gives a first and brief introduction to ergodic theory, avoiding on purpose more advanced topics. After introducing the notions of a measurable map and of an invariant measure, we establish Poincaré’s recurrence theorem and Birkhoff’s ergodic theorem. We also consider briefly the notions of Lyapunov exponent and of metric entropy. The pre-requisites from measure theory and integration theory are fully recalled in Sect. 8.1.
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- 1.
Theorem (See for example [27]) Given a measure space (X,μ), if \(\varphi_{n} \colon X \to \mathbb{R}_{0}^{+}\) is a sequence of measurable functions, then
$$\int_X \liminf_{n \to\infty} \varphi_n \, d \mu \le\liminf_{n \to\infty} \int_X \varphi_n \, d \mu. $$ - 2.
Theorem (See for example [27]) Given a measure space (X,μ), if \(\varphi_{n} \colon X \to \mathbb{R}_{0}^{+}\) is a nondecreasing sequence of measurable functions, then
$$\int_X \lim_{n \to\infty} \varphi_n \, d \mu= \lim_{n \to\infty} \int_X \varphi_n \, d \mu. $$
References
Friedman, A.: Foundations of Modern Analysis. Dover, New York (1982)
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© 2013 Springer-Verlag London
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Barreira, L., Valls, C. (2013). Ergodic Theory. In: Dynamical Systems. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4835-7_8
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DOI: https://doi.org/10.1007/978-1-4471-4835-7_8
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