Let X be a set and T: X ~» X be a bi jection. Consider the space ^ of pseudo-Menger maps on X whi... more Let X be a set and T: X ~» X be a bi jection. Consider the space ^ of pseudo-Menger maps on X which induce a compact topology on X for which T is a homeomorphism. The lattice properties of Λί are investigated and a bivariate nonnegative function of-^ is defined which possesses certain properties analogous to those of the usual conditional entropy function defined on the space of measurable partitions of a probability space.
A pseudo-Menger space is a set X together with a function ^:IxI-»S, the set of distribution funct... more A pseudo-Menger space is a set X together with a function ^:IxI-»S, the set of distribution functions, satisfying certain natural axioms similar to those of a pseudo-metric space. Let T: J-> J be a bijection and let θ τ denote the topology generated by {TWip, e, λ): i e Z, p6 X, ε > 0, λ > 0} where U(p, ε, λ) = {q: θ(p, q)(ε) > 1-λ}. Assume that θ τ is compact. Let h(T,θ) denote the topological entropy of T with respect to the θ τ topology. The purpose of this note is to show that if one is given a sequence {#"} of pseudo-Menger structures on X satisfying θ n (p, q) Ξ> θ(p, q) and θ n (p, q)-» θ(p, q) in distribution for all p,qeX then h(T,θ n)->h(T 9 θ). A counterexample is then given to show that, in general, the condition θ n (p, q) i> θ{p, q) cannot be removed.
1. Introduction. The following discussion is an application of Ornstein's recent results concerni... more 1. Introduction. The following discussion is an application of Ornstein's recent results concerning Bernoulli automorphisms to the study of induced transformations. Let (Y, ~), tz) be a non-atomic Lebesgue space and let T be an automorphism of X. If A is a measurable subset of X and / /°° T~A wi=o = X, then T induces on the Lebesgue space A with the conditional measure t~a the induced automorphism Ta defined by the formula TA(x)= Tk(x) (x c A), where k = k(x) is the smallest positive integer j such that Ti(x) ~ A. It will be shown that for a certain class of subsets of a Bernoulli shift weak mixing is a sufficient condition to ensure that the induced transformation be Bernoulli. As an application, one deduces that any two Bernoulli shifts are equivalent in the sense of Kakutani; that is, they can be imbedded as induced transformations in a single ergodic system of finite measure. Conversely, given a weak mixing system (Y, S), we ask what knowledge can be derived about S knowing the existence of a subset A of Y for which SA is Bernoulli. More precisely, it is shown that if co is the partition of A into return times of T, B a Bernoulli partition for (A, Ta), and H~,A(oJ+/B +) < oo (where + denotes the "future" of a partition under Ta), then S is weakly mixing if and only if S is a Bernoulli shift. 2. Preliminaries. We will adhere to the definitions and notation of entropy theory established in [11]. If ~ is a measurable partition of (X, ~), then o(~) will denote the sub<r-algebra consisting of those M-measurable sets which are unions of elements of ~. The infinite partitions V~T i~ and V-~T i~ will be denoted by ~+ and ~-respectively. If G is a finite generator of the or-algebra of X such that {TiG:-oo < i < ~ } are jointly independent, then G will be called a Bernoulli partition for T. Given finite partitions P and Q of X and E > 0 we will write P_[_'~Q if and only if Zi[~(Pi/QJ)-u(Pi)] < ~ for all but a set of atoms of Q whose union * These results are contained in the author's doctoral dissertation written at the University of California at Berkeley under the guidance of Professor Jacob Feldman to whom the author is very grateful for both enc,~uragement and advice. In addition, the author wishes to thank Professor William Parr,' for several helpful suggestions.
The purpose of this note is to study a certain class of stopping times for Bernoulli automorphism... more The purpose of this note is to study a certain class of stopping times for Bernoulli automorphisms by means of the Friedman-Ornstein results concerning weakly Bernoulli partitions.
1. Introduction. The following discussion is an application of Ornstein's recent results concerni... more 1. Introduction. The following discussion is an application of Ornstein's recent results concerning Bernoulli automorphisms to the study of induced transformations. Let (Y, ~), tz) be a non-atomic Lebesgue space and let T be an automorphism of X. If A is a measurable subset of X and / /°° T~A wi=o = X, then T induces on the Lebesgue space A with the conditional measure t~a the induced automorphism Ta defined by the formula TA(x)= Tk(x) (x c A), where k = k(x) is the smallest positive integer j such that Ti(x) ~ A. It will be shown that for a certain class of subsets of a Bernoulli shift weak mixing is a sufficient condition to ensure that the induced transformation be Bernoulli. As an application, one deduces that any two Bernoulli shifts are equivalent in the sense of Kakutani; that is, they can be imbedded as induced transformations in a single ergodic system of finite measure. Conversely, given a weak mixing system (Y, S), we ask what knowledge can be derived about S knowing the existence of a subset A of Y for which SA is Bernoulli. More precisely, it is shown that if co is the partition of A into return times of T, B a Bernoulli partition for (A, Ta), and H~,A(oJ+/B +) < oo (where + denotes the "future" of a partition under Ta), then S is weakly mixing if and only if S is a Bernoulli shift. 2. Preliminaries. We will adhere to the definitions and notation of entropy theory established in [11]. If ~ is a measurable partition of (X, ~), then o(~) will denote the sub<r-algebra consisting of those M-measurable sets which are unions of elements of ~. The infinite partitions V~T i~ and V-~T i~ will be denoted by ~+ and ~-respectively. If G is a finite generator of the or-algebra of X such that {TiG:-oo < i < ~ } are jointly independent, then G will be called a Bernoulli partition for T. Given finite partitions P and Q of X and E > 0 we will write P_[_'~Q if and only if Zi[~(Pi/QJ)-u(Pi)] < ~ for all but a set of atoms of Q whose union * These results are contained in the author's doctoral dissertation written at the University of California at Berkeley under the guidance of Professor Jacob Feldman to whom the author is very grateful for both enc,~uragement and advice. In addition, the author wishes to thank Professor William Parr,' for several helpful suggestions.
A pseudo-Menger space is a set X together with a function ^:IxI-»S, the set of distribution funct... more A pseudo-Menger space is a set X together with a function ^:IxI-»S, the set of distribution functions, satisfying certain natural axioms similar to those of a pseudo-metric space. Let T: J-> J be a bijection and let θ τ denote the topology generated by {TWip, e, λ): i e Z, p6 X, ε > 0, λ > 0} where U(p, ε, λ) = {q: θ(p, q)(ε) > 1-λ}. Assume that θ τ is compact. Let h(T,θ) denote the topological entropy of T with respect to the θ τ topology. The purpose of this note is to show that if one is given a sequence {#"} of pseudo-Menger structures on X satisfying θ n (p, q) Ξ> θ(p, q) and θ n (p, q)-» θ(p, q) in distribution for all p,qeX then h(T,θ n)->h(T 9 θ). A counterexample is then given to show that, in general, the condition θ n (p, q) i> θ{p, q) cannot be removed.
Let X be a set and T: X ~» X be a bi jection. Consider the space ^ of pseudo-Menger maps on X whi... more Let X be a set and T: X ~» X be a bi jection. Consider the space ^ of pseudo-Menger maps on X which induce a compact topology on X for which T is a homeomorphism. The lattice properties of Λί are investigated and a bivariate nonnegative function of-^ is defined which possesses certain properties analogous to those of the usual conditional entropy function defined on the space of measurable partitions of a probability space.
The purpose of this note is to study a certain class of stopping times for Bernoulli automorphism... more The purpose of this note is to study a certain class of stopping times for Bernoulli automorphisms by means of the Friedman-Ornstein results concerning weakly Bernoulli partitions.
Journal of Mathematical Analysis and Applications, 1977
The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a L... more The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a Lebesgue space X, 9s denote the set of all partitions of X possessing finite entropy, and 9' denote the set of all increasing sequences of positive integers.
Journal of Mathematical Analysis and Applications, 1977
The purpose of this note is to provide a new proof to the following reformulation of the Suchesto... more The purpose of this note is to provide a new proof to the following reformulation of the Sucheston zero-one law: An automorphism T of a Lebesgue space X is mixing if and only if, for every subsequence A of the sequence of natural numbers and every partition a of X having finite entropy, there exists a subsequence B = {b(j)} of A such that A,zl V,?, T*lj)(ar) is the trivial partition.
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Papers by Alan J Saleski