Abstract
In this chapter we will describe a topology on the class of non-singular automorphisms on a measure space and discuss the Baire category of some naturally occurring subclasses of it. We shall make a similar study of the class of measure preserving automorphisms.
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References
S. Banach. Théorie des Opérations Linéaires, Chelsea, New York, 1963.
G. D. Birkhoff. Probability and Physical Systems, Bull. Amer. Math. Soc. 138 (1932), 361–379. Birkhoff: Collected Mathematical Papers, vol. 2.
J. R. Choksi and S. Kakutani. Residuality of Ergodic Measurable Transformations and Transformations which Preserve an Infinite Measure, Indiana University Mathematics Journal, 128 (1979), 453–469.
J. R. Choksi and M. G. Nadkarni. Baire Category in Spaces of Measures, Unitary Operators and Transformations, Proceedings of the International Conference on Invariant Subspaces and Allied Topics, University of Delhi, H. Helson and B. S. Yadav, Editors, Narosa Publishers, New Delhi.
I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai. Ergodic Theory, Springer-Verlag, New York, (1981).
A. del Junco. Disjointness of Measure Preserving Transformations, Minimal Self Joinings and Category, Ergodic Theory and Dynamical Systems I, Progress in Mathematics 10, Birkhauser, Boston, 1981, 81–89.
H. Furstenberg. Disjointness in Ergodic Theory, Minimal Sets, and a Problem in Diophantine Approximation, Math. Systems Theory, (1967) 1–50.
N. Friedman. Introduction to Ergodic Theory, van Nostrand-Reinhold, New York, 1970.
F. Hahn and W. Parry. Some Characteristic Properties of Dynamical Systems with Quasi-Discrete Spectrum, Math. Systems Theory, 2 (1968), 179–190.
P. R. Halmos. Lectures on Ergodic Theory, Math. Soc. Japan Publication, Tokyo 1956. Reprinted Chelsea New York, 1960.
P. R. Halmos. Approximation Theories for Measure Preserving Transformations, Trans. Amer. Math. Soc. 55 (1944) 1–18.
P. R. Halmos. In General a Measure Preserving Transformation is mixing, Ann. of Math. 45 (1944), 786–792.
J. M. Hawkins and E. A. Robinson Jr. Approximately Transitive Flows and Transformations Have Simple Spectrum, Preprint 1985.
A. Katok and A. M. Stepin. Approximation in Ergodic Theory, Russian. Math. Surveys, 22 (1967), 77–102.
A. Katok. Approximation and Genericity in Abstract Ergodic Theory, Notes 1985.
J. C. Oxtoby and S. Ulam. Measure Preserving Homeomorphisms and Metric Transitivity, Ann. Math. (2) 42 (1941), 874–920.
K. Petersen. Ergodic Theory, Cambridge Studies in Advanced Mathematics; 2, Cambridge University Press, 1983.
V. A. Rokhlin. New Progress in the Theory of Transformations with Invariant Measure, Russian. Math. Surveys, 15 (1960), 1–22.
V. A. Rokhlin. The General Measure Preserving Transformation is Mixing, Dokl. Acad. Sci. USSR, 3 (1948), 349–358.
H. L. Royden. Real Analysis, Edition 3, MacMillan Publishing Co., (1989), New York.
B. Simon. Operators with Singular Continuous Spectrum: I. General Operators, Annals of Math. 141 (1995), 131–145.
A. M. Stepin. Spectral Properties of Generic Dynamical Systems, Math. USSR Izvestiya, 29 (1987), No.1.
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© 1998 Hindustan Book Agency
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Nadkarni, M.G. (1998). Baire Category Theorems of Ergodic Theory. In: Spectral Theory of Dynamical Systems. Texts and Readings in Mathematics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-80250-93-9_8
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DOI: https://doi.org/10.1007/978-93-80250-93-9_8
Publisher Name: Hindustan Book Agency, Gurgaon
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