Dimensionally preserving figure
Dimensionally preserving maps , sometimes also called true to size maps , are self- maps of a probability space that receive the probability measure . One also speaks of dimensionally conserving dynamic systems , especially when one considers the behavior of the mapping under iteration .
Dimensional preserving images are the subject of ergodic theory within the theory of dynamic systems .
definition
Let be a probability space , i.e. i.e., be a set, the σ-algebra of measurable sets, and a probability measure . A measurable figure
means true-to- size mapping, if for everyone
applies.
It should be noted that for a dimensionally preserving mapping it does not necessarily have to apply to the measurable quantities , i.e. that only archetypes and not necessarily images of measurable quantities have the same measure. The picture on the right shows the Bernoulli mapping (angle doubling) . This mapping is dimensionally conservative, for example applies to every interval , that is
- .
Nevertheless, image sets do not have to have the same measure as the original set, for example is , but .
Examples
Dimensionally preserving illustrations
- be the unit circle , the σ-algebra of the Borel sets and the uniformly distributed probability measure . Every rotation of the unit circle is a dimensionally preserving image.
- The self-mapping of the n-dimensional torus , which is defined by an integer, unimodular matrix , is dimensionally preserving with regard to the probability measure .
- An interval exchange mapping is dimensionally conservative.
Dimensionally preserving dynamic systems
The stationary stochastic processes in discrete time form an important class of dimensionally maintaining dynamic systems . To do this, one defines a canonical process and the shift operator as
- .
Then there is and is a dynamic system that is dimensionally preserving due to the stationarity.
Invariants
An invariant that measures the chaotic nature of scale-preserving images is the Kolmogorow-Sinai entropy .
literature
- Peter Walters: Ergodic theory — introductory lectures (= Lecture Notes in Mathematics. Vol. 458). Springer, Berlin / New York, 1975.
- James R. Brown: Ergodic theory and topological dynamics (= Pure and Applied Mathematics. 70). Academic Press [Harcourt Brace Jovanovich, Publishers], New York / London, 1976.
- H. Furstenberg: Recurrence in ergodic theory and combinatorial number theory. (= MB Porter Lectures ). Princeton University Press, Princeton, NJ, 1981, ISBN 0-691-08269-3 .
- Daniel J. Rudolph: Fundamentals of Measurable Dynamics. Ergodic theory on Lebesgue spaces. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990, ISBN 0-19-853572-4 .
- Ya. G. Sinaĭ: Topics in ergodic theory (= Princeton Mathematical Series. 44). Princeton University Press, Princeton, NJ, 1994, ISBN 0-691-03277-7 .
- CE Silva: Invitation to ergodic theory (= Student Mathematical Library. 42). American Mathematical Society, Providence, RI, 2008, ISBN 978-0-8218-4420-5 .
- Alexander S. Kechris: Global aspects of ergodic group actions (= Mathematical Surveys and Monographs. 160). American Mathematical Society, Providence, RI, 2010, ISBN 978-0-8218-4894-4 .
- Steven Kalikow, Randall McCutcheon: An outline of ergodic theory (= Cambridge Studies in Advanced Mathematics. 122). Cambridge University Press, Cambridge 2010, ISBN 978-0-521-19440-2 .