Chapter 38: Automorphisms
381
Automorphisms of Boolean algebras
Assembling an automorphism; elements supporting an automorphism;
periodic and aperiodic parts; full and countably full subgroups;
recurrence; induced automorphisms of principal ideals; Stone spaces;
cyclic automorphisms.
382
Factorization of automorphisms
Separators and transversals; Frolik's theorem; exchanging
involutions; expressing an automorphism as the product of three
involutions; subgroups of Aut(A) with many involutions;
normal subgroups of full groups with many involutions; simple
automorphism groups.
383
Automorphism groups of measure algebras
Measure-preserving automorphisms as products of involutions; normal
subgroups of Aut(A) and Autμ(A); conjugacy
in Aut(A) and Autμ(A).
384
Outer automorphisms
If G≤Aut(A), H≤Aut(B) have many involutions, any
isomorphism between G and H arises from an isomorphism between
A and B; if A is nowhere rigid,
Aut(A) has no outer automorphisms; applications to
localizable measure algebras.
385
Entropy
Entropy of a partition of unity in a probability algebra; conditional
entropy; entropy of a measure-preserving homomorphism; calculation
of entropy (Kolmogorov-Sinai theorem); Bernoulli shifts;
isomorphic homomorphisms and conjugacy classes in
Autμ(A); almost isomorphic inverse-measure-preserving functions.
386
More about entropy
The Halmos-Rokhlin-Kakutani lemma; the Shannon-McMillan-Breiman
theorem; the Csiszár-Kullback inequality; various lemmas.
387
Ornstein's theorem
Bernoulli partitions; finding Bernoulli partitions with elements of
given measure (Sinai's theorem); adjusting Bernoulli
partitions; Ornstein's theorem (Bernoulli shifts of the same finite
entropy are isomorphic); Ornstein's and Sinai's theorems in the
case of infinite entropy.
388
Dye's theorem
Orbits of inverse-measure-preserving functions; von Neumann transformations; von Neumann
transformations generating a given full subgroup; classification of
full subgroups generated by a single automorphism.
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