Abstract
Given a Borel automorphism σ on a Borel space (X, Ɓ) and a set A in Ɓ, the Poincaré recurrence lemma permits us to define (mod W σ ) a natural automorphism on A, where W σ denotes the σ ideal generated by the class of sets in Ɓ wandering under σ,(See Chapter 1). This map was given the name “induced transformation” (“induced automorphism”) by S. Kakutani who also studied its properties and used it to define a new kind of equivalence among the measure preserving automorphisms, now called Kakutani equivalence. In our exposition below of these concepts we will partly follow N. Friedman [2] who made these and related ideas available to a wider public.
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Bibliography
G. D. Birkhoff. Proof of a Recurrence Theorem for Strongly Transitive Systems, Proc. Nat. Acad. Sci., U. S. A., 17 (1931), 650–655. Birkhoff: Collected Mathematical Papers, vol. 2, 398–408, Dover, 1968.
N. A. Friedman. Introduction to Ergodic Theory, Van Nostrand-Reinhold, 1970.
P. Halmos. Approximation Theories for Measure Preserving Transformations, Trans. Amer. Math. Society, 55 (1944), 1–18.
Mark Kac. On the Notion of Recurrence in Discrete Stochastic Processes, Bull. Amer. Math. Soc. 53 (1947), 1002–1010.
S. Kakutani. Induced Measure Preserving Transformations, Proc. Imp. Acad. Tokyo, 19 (1943), 635–641.
A. Kechris. Lectures on Definable Group Actions and Equivalence Relations, Lecture Notes, Caltech, Pasadena, 1993.
D. S. Ornstein, D. J. Rudolph, B. Weiss. Equivalence of Measure Preserving Transformations, Memoirs of the Amer. Math. Soc., vol. 37, No. 262, 1982.
V. A. Rokhlin. Selected Problems in The Metric Theory of Dynamical Systems, Uspehi. Math. Nauk. 30 (1949), 57–108.
V. A. Rokhlin. Generators in Ergodic Theory I, Vestnik Leningrad Univ. Math., (1963), 26–32.
B. Weiss. Measurable Dynamics, Contemporary Math. 26 Amer. Math. Soc., (1984), 397–421.
B. Weiss. Countable Generators in Dynamics-Universal Minimal Models, Contemporary Math., 94 Amer. Math. Soc., (1989), 321–326.
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© 2013 Hindustan Book Agency
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Nadkarni, M.G. (2013). Induced Automorphisms and Related Concepts. In: Basic Ergodic Theory. Texts and Readings in Mathematics, vol 6. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-53-8_7
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DOI: https://doi.org/10.1007/978-93-86279-53-8_7
Publisher Name: Hindustan Book Agency, Gurgaon
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