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 ${\displaystyle (X,\Sigma ,P)}$ ${\displaystyle X}$ ${\displaystyle \Sigma \subset {\mathcal {P}}(X)}$ ${\displaystyle P}$

T  : [0,1) → [0,1), is a measure-preserving mapping for the Lebesgue measure on [0,1]. ${\displaystyle x\mapsto 2x\mod 1}$

${\displaystyle f\colon X\to X}$

means true-to- size mapping, if for everyone ${\displaystyle A\in \Sigma }$

${\displaystyle P(f^{-1}(A))=P(A)}$

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 ${\displaystyle P(f(B))=P(B)}$ ${\displaystyle B}$ ${\displaystyle T(x)=2x\mod 1}$ ${\displaystyle T^{-1}(a,b)=({\frac {a}{2}},{\frac {b}{2}})\cup ({\frac {a+1}{2}},{\frac {b+1}{2}})}$

${\displaystyle P(T^{-1}(a,b))={\frac {b-a}{2}}+{\frac {b-a}{2}}=b-a=P(a,b)}$.

Nevertheless, image sets do not have to have the same measure as the original set, for example is , but . ${\displaystyle P(0,{\frac {1}{2}})=0{,}5}$ ${\displaystyle P(T(0,{\frac {1}{2}}))=P(0,1)=1}$

Examples

Dimensionally preserving illustrations

• ${\displaystyle X}$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. ${\displaystyle \Sigma }$ ${\displaystyle P}$ ${\displaystyle {\frac {1}{2\pi }}d\theta }$
• 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 . ${\displaystyle A\in GL(n,\mathbb {Z} )}$ ${\displaystyle f\colon T^{n}\to T^{n}}$ ${\displaystyle T^{n}=\mathbb {R} ^{n}/\mathbb {Z} ^{n}}$ ${\displaystyle {\frac {1}{(2\pi )^{n}}}d\theta _{1}\ldots d\theta _{n}}$
• 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 ${\displaystyle (\Omega ,{\mathcal {A}},P)=(E^{\times \mathbb {N} },{\mathcal {B}}(E)^{\otimes E},P)}$ ${\displaystyle \tau }$

${\displaystyle \tau ((\omega _{n})_{n\in \mathbb {N} })=(\omega _{n+1})_{n\in \mathbb {N} }}$.

Then there is and is a dynamic system that is dimensionally preserving due to the stationarity. ${\displaystyle X_{n}(\omega )=X_{0}(\tau ^{n}(\omega ))}$ ${\displaystyle (\Omega ,{\mathcal {A}},P,\tau )}$

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 .