The asymptotic distributional behaviour of transformations preserving infinite measures

A problem of equidistribution on the unit interval [0, 1]

We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a two-symbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing.

Dynamical properties of the Pascal adic transformation

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures---a version of de Finetti's Theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

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Symmetric Gibbs measures

We construct funny rank-one infinite measure preserving free actionsT of a countable Abelian groupG satisfying each of the following properties: (1)T
 g1×…×Tgk is ergodic for each finite sequenceg
 1,…,g
 k ofG-elements of infinite order, (2)T×T is nonconservative, (3)T×T is nonergodic but allk-fold Cartesian products are conservative, and theL
 ∞-spectrum ofT is trivial, (4) for eachg of infinite order, allk-fold Cartesian products ofT
 g are ergodic, butT
 2g×Tg is nonconservative. A topological version of this theorem holds. Moreover, given an AT-flowW, we construct nonsingularG-actionsT with similar properties and such that the associated flow ofT isW. Orbit theory is used in an essential way here.

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Funny rank-one weak mixing for nonsingular abelian actions

Invariant measures and orbits of dissipative transformations

Full groups and a theorem of dye

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Ergodic Measure Preserving Transformations of Finite Type

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Iterates of measurable transformations and Markov operators

In the following we shall consider only nonatomic, a-finite measure spaces (X, %, m). We say that a measure /i is equivalent to m if /i and m have the same sets of measure zero. We shall discuss measurable transformations T that are 1-1 onto maps with measurable inverses. If m(TA) = m(A) for all sets A G % we say that T is a measure preserving transformation, or that m is an invariant measure for the transformation T. We assume that all transformations mentioned are nonsingular; in other words, the image of a set of positive measure has positive measure also. A measurable transformation T is ergodic if TA = A implies that either the set A or its complement has measure zero. We shall often tacitly assume the phrase \"almost everywhere\", and all sets considered shall be measurable. In [13] E. Hopf first discussed a necessary and sufficient condition for the existence of a finite invariant measure /x equivalent to m. Since then many authors have discussed different aspects of transformations without a finite invariant measure and have obtained a number of interesting results that have deep connections with other areas of mathematics. For instance, there is a significant influence on the classification theory of the factors of a von Neumann algebra. In [10] weakly wandering sets for a transformation T were introduced, and it was shown that there exists a finite measure /i equivalent to m and invariant for T if and only if there are no weakly wandering sets of positive measure for the transformation T. Let % = {nt\\i = 0, 1, 2 , . . . } be a sequence of integers; we say that a set A is a weakly wandering set under the sequence % for the transformation T if the sets TA f or i = 0, 1, 2 , . . . are mutually disjoint. If the weakly wandering set A has positive measure then we say that % is a weakly wandering sequence for the transformation T. In case the sequence ^ consists of the set of all integers then we have the familiar case where A is a wandering set. Recurrent transformations play an important role in ergodic theory; these are the transformations which do not accept wandering sets of positive measure. Strongly recurrent transformations were introduced and discussed in [3]; these are the transformations which do not accept weakly wandering sets of positive measure. An ergodic measure preserving transformation T is

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Yuji Ito

Papers

Invariant measures and orbits of dissipative transformations

Full groups and a theorem of dye

Ergodic Measure Preserving Transformations of Finite Type

Iterates of measurable transformations and Markov operators

Transformations that do not accept a finite invariant measure