Kakutani–Rokhlin towers
E665942
Kakutani–Rokhlin towers are combinatorial structures in ergodic theory that decompose a measure-preserving transformation into stacked levels (or “towers”) to analyze its dynamical and measure-theoretic properties.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kakutani–Rokhlin towers canonical | 1 |
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
construction in ergodic theory
ⓘ
mathematical concept ⓘ |
| appliesTo |
invertible measure-preserving transformations
ⓘ
non-invertible measure-preserving transformations ⓘ |
| associatedWith | measure-preserving transformation ⓘ |
| category | combinatorial structure in ergodic theory ⓘ |
| definedOver |
measure space
ⓘ
probability space ⓘ |
| describedAs |
decomposition of space into stacked levels along orbits
ⓘ
tower decomposition of a measure-preserving system ⓘ |
| field |
dynamical systems
ⓘ
ergodic theory ⓘ measure theory ⓘ |
| hasPart |
base set
ⓘ
columns ⓘ heights of towers ⓘ levels ⓘ |
| namedAfter |
Shizuo Kakutani
NERFINISHED
ⓘ
Vladimir Rokhlin NERFINISHED ⓘ |
| property |
allow approximation of transformation by cyclic permutations on levels
ⓘ
bases can be chosen with arbitrarily small measure ⓘ heights can be chosen large to approximate long orbit segments ⓘ levels are images of the base under iterates of the transformation ⓘ levels in a tower are pairwise disjoint ⓘ provide combinatorial model of the dynamics ⓘ union of towers covers most of the space up to small measure error ⓘ |
| relatedTo |
Kakutani skyscraper construction
ⓘ
Rokhlin lemma NERFINISHED ⓘ induced transformations ⓘ symbolic dynamics ⓘ |
| usedFor |
analysis of entropy
ⓘ
approximation of invariant measures ⓘ classification of measure-preserving transformations up to isomorphism ⓘ construction of Markov partitions in some settings ⓘ construction of factors of dynamical systems ⓘ construction of rank-one transformations ⓘ proofs of mixing and weak mixing properties ⓘ |
| usedIn |
approximation of dynamical systems by periodic processes
ⓘ
construction of Kakutani skyscrapers ⓘ construction of generating partitions ⓘ ergodic decomposition techniques ⓘ orbit decomposition ⓘ proofs of ergodic theorems ⓘ study of measure-preserving transformations ⓘ symbolic coding of dynamical systems ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.