Inverse limits, entropy and weak isomorphism for discrete dynamical systems
Author:
James R. Brown
Journal:
Trans. Amer. Math. Soc. 164 (1972), 5566
MSC:
Primary 28A65
MathSciNet review:
0296251
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Abstract: A categorical approach is taken to the study of a single measurepreserving transformation of a finite measure space and to inverse systems and inverse limits of such transformations. The questions of existence and uniqueness of inverse limits are settled. Sinai's theorem on generators is recast and slightly extended to say that entropy respects inverse limits, and various known results about entropy are obtained as immediate corollaries, e.g. systems with quasidiscrete or quasiperiodic spectrum have zero entropy. The inverse limit of an inverse system of dynamical systems is (1) ergodic, (2) weakly mixing, (3) mixing (of any order) iff each has the same property. Finally, inverse limits are used to lift a weak isomorphism of dynamical systems and to an isomorphism of systems and with the same entropy.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202962517
PII:
S 00029947(1972)02962517
Keywords:
Inverse limits,
dynamical systems,
measurepreserving transformation,
factor,
invariant subalgebra,
weakly isomorphic,
direct product,
bounded inverse system,
Lebesgue system,
discrete spectrum,
exact system,
natural extension,
disjoint,
ergodic,
weakly mixing,
mixing (of any order),
quasidiscrete spectrum,
quasiperiodic spectrum
Article copyright:
© Copyright 1972
American Mathematical Society
