Kakutani skyscraper construction

E665941

construction in ergodic theory method in dynamical systems

Kakutani skyscraper construction is a method in ergodic theory for building measure-preserving transformations by stacking intervals into “skyscrapers” to analyze and classify dynamical systems.

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Statements (44)

Predicate	Object
instanceOf	construction in ergodic theory ⓘ method in dynamical systems ⓘ
appliedIn	classification of measure-preserving transformations ⓘ ergodic decomposition problems ⓘ symbolic dynamics via tower representations ⓘ
assumes	measurable transformation ⓘ sigma-finite measure space ⓘ
constructionStep	define a transformation that moves points up the tower ⓘ define return times to the base set ⓘ identify the top of each column with the base via the induced map ⓘ partition a base set into subintervals or subsets ⓘ stack copies of the base according to return times ⓘ
domain	probability space ⓘ standard measure space ⓘ
field	dynamical systems ⓘ ergodic theory ⓘ measure theory ⓘ
historicalContext	introduced in mid-20th century ergodic theory ⓘ
mathematicalArea	analysis ⓘ probability theory on dynamical systems ⓘ pure mathematics ⓘ
namedAfter	Shizuo Kakutani NERFINISHED ⓘ
output	measure-preserving transformation on a tower ⓘ skyscraper representation of a transformation ⓘ
property	preserves the underlying measure by construction ⓘ represents transformations as stacks of intervals or sets ⓘ
purpose	to analyze dynamical systems ⓘ to build measure-preserving transformations ⓘ to classify measure-preserving transformations ⓘ
relatedTo	Kakutani equivalence NERFINISHED ⓘ Rokhlin lemma NERFINISHED ⓘ cutting and stacking construction ⓘ induced transformations in ergodic theory ⓘ
usedFor	comparing measure-preserving systems up to Kakutani equivalence ⓘ constructing examples of ergodic transformations ⓘ constructing examples of non-isomorphic but Kakutani equivalent systems ⓘ studying orbit structure of transformations ⓘ
usesConcept	Poincaré recurrence NERFINISHED ⓘ Rokhlin tower NERFINISHED ⓘ induced transformation ⓘ interval partition ⓘ measure-preserving transformation ⓘ return time ⓘ tower decomposition ⓘ

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Kakutani equivalence in ergodic theory → relatedTo → Kakutani skyscraper construction ⓘ