Some invariants of Kakutani equivalence

1981 ◽  
Vol 38 (3) ◽  
pp. 231-240 ◽  
Author(s):  
Marina Ratner
Keyword(s):  
Kakutani Equivalence

scholarly journals Descriptive Kakutani equivalence

2010 ◽  
pp. 179-219 ◽  
Author(s):  
Benjamin Miller ◽  
Christian Rosendal
Keyword(s):  
Kakutani Equivalence

Flow–orbit equivalence for minimal Cantor systems

2008 ◽  
Vol 28 (2) ◽  
pp. 481-500 ◽  
Author(s):  
WOJCIECH KOSEK ◽  
NICHOLAS ORMES ◽  
DANIEL J. RUDOLPH
Keyword(s):  
Distortion Function ◽  
Full Group ◽  
Orbit Equivalence ◽  
Basic Facts ◽  
The Relationship ◽  
Kakutani Equivalence

AbstractThis paper is about flow–orbit equivalence, a topological analog of even Kakutani equivalence. In addition to establishing many basic facts about this relation, we characterize the conjugacies of induced systems that can be extended to a flow–orbit equivalence. We also describe the relationship between flow–orbit equivalence and a distortion function of an orbit equivalence. We show that, if the distortion of an orbit equivalence is zero, then it is in fact a flow–orbit equivalence, and that the converse is true up to a conjugation by an element of the full group.

scholarly journals Kakutani equivalence for products of some special flows over rotations

2021 ◽  
pp. 1-28
Author(s):  
DAREN WEI
Keyword(s):  
Full Measure ◽  
Countable Family ◽  
Function Smooth ◽  
Roof Function ◽  
Kakutani Equivalence

Abstract We study Kakutani equivalence for products of some special flows over rotations with roof function smooth except a singularity at $0\in \mathbb {T}$ . We estimate the Kakutani invariant for products of these flows with different powers of singularities and rotations from a full measure set. As a corollary, we obtain a countable family of pairwise non-Kakutani equivalent products of special flows over rotations.

scholarly journals Nearly continuous Kakutani equivalence of adding machines

2009 ◽  
Vol 3 (1) ◽  
pp. 103-119 ◽  
Author(s):  
Mrinal Kanti Roychowdhury ◽  
◽  
Daniel J. Rudolph ◽  
Keyword(s):  
Nearly Continuous ◽  
Adding Machines ◽  
Kakutani Equivalence

α-equivalence: a refinement of Kakutani equivalence

1994 ◽  
Vol 14 (1) ◽  
pp. 69-102 ◽  
Author(s):  
Adam Fieldsteel ◽  
Andrés Del Junco ◽  
Daniel J. Rudolph
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Point Spectrum ◽  
Return Time ◽  
Equivalence Theorem ◽  
Weakly Mixing ◽  
Bernoulli Flow ◽  
Measure Preserving Transformations ◽  
Kakutani Equivalence

AbstractFor a fixed irrational α > 0 we say that probability measure-preserving transformationsSandTare α-equivalent if they can be realized as cross-sections in a common flow such that the return time functions on the cross-sections both take values in {1, 1 +α} and have equal integrals. Similarly we call two flowsFandGα-equivalent ifFhas a cross-sectionSandGhas a cross-sectionTisomorphic toSand again both the return time functions take values in {1, 1 + α} and have equal integrals. The integer kα(S), equal to the least positivesuchsuch that exp2πikα-1belongs to the point spectrum ofS, is an invariant of α-equivalence.We obtain a characterization of a-equivalence as a particular type of restricted orbit equivalence and use this to prove that within the class of loosely Bernoulli mapska(S) together with the entropyh(S) are complete invariants of α-equivalence. There is a corresponding a-equivalence theorem for flows which has as a consequence, for example, that up to an obvious entropy restriction, any weakly mixing cross-section of a loosely Bernoulli flow can also be realized as a cross-section with a {1,1 + α}-valued return time function.For the proof of the α-equivalence theorem we develop a relative Kakutani equivalence theorem for compact group extensions which is of interest in its own right. Finally, an example of Fieldsteel and Rudolph is used to show that in generalkα(S) is not a complete invariant of α-equivalence within a given even Kakutani equivalence class.

scholarly journals Kakutani equivalence of ergodic ℤn actions

1984 ◽  
Vol 4 (1) ◽  
pp. 89-104 ◽  
Author(s):  
Andres Del Junco ◽  
Daniel J. Rudolph
Keyword(s):  
Classical Theory ◽  
Cross Sections ◽  
One Dimension ◽  
Asymptotic Linearity ◽  
Natural Way ◽  
Linearity Condition ◽  
Kakutani Equivalence

AbstractWe define two families of relations between ergodic ℤn actions, both indexed equivariantly by non-singular n × n matrices. The first is to be Katok cross-sections of the same flow, indexed in a natural way by the matrices. The second is determined by the existence of an orbit preserving injection with an extra asymptotic linearity condition. We demonstrate that these two families are identical. In one dimension this is the classical theory of Kakutani equivalence.

scholarly journals Measured topological orbit and Kakutani equivalence

2009 ◽  
Vol 2 (2) ◽  
pp. 221-238 ◽  
Author(s):  
Andres del Junco ◽  
◽  
Daniel J. Rudolph ◽  
Benjamin Weiss ◽  
◽  
...  
Keyword(s):  
Kakutani Equivalence
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