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 Aut_{μ}(**A**); conjugacy
in Aut(**A**) and Aut_{μ}(**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
Aut_{μ}(**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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Revised 7.6.2016