Cambridge University Press eBooks· 2023· Book chapter
Geometric Measure Theory
- 344citations
- 2023year
Short summary
This chapter introduces fundamental tools for refined ergodic theory, including quasi-invariant measures, first return maps, and nice sets, culminating in proofs of key ergodic theorems.
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Key points
- Introduces quasi-invariant measures and the first return map as fundamental tools.
- Proves the Poincaré Recurrence Theorem, Birkhoff Ergodic Theorem, and Hopf Ergodic Theorem.
- Presents a method for finding invariant measures absolutely continuous with respect to quasi-invariant measures.
AI-generated from the title and abstract; the full text is not read.
Abstract
We start with quasi-invariant measures and early on, in the second section of this chapter, we introduce the powerful concept of the first return map. This concept, along with the concept of nice sets, forms our most fundamental tool in Part IV of our book, which is devoted to presenting a refined ergodic theory of elliptic functions. We introduce, in this chapter, the notions of ergodicity and conservativity (always satisfied for finite invariant measures), and prove the Poincaré Recurrence Theorem, Birkhoff Ergodic Theorem, and Hopf Ergodic Theorem, the last pertaining to infinite measures. We also provide a powerful, though perhaps somewhat neglected by the ergodic community, tool for proving the existence of invariant s-finite measures absolutely continuous with respect to given quasi-invariant measures.
The authors' abstract, as published at the source. Cambridge University Press eBooks, 2023 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics