Vidhu Prasad

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• Ph. D.
• Professor (Full) at University of Massachusetts Lowell

This article is concerned with demonstrating the power and simplicity of sww (special weakly wandering) sequences. We calculate an sww growth sequence for the infinite measure preserving random walk transformation. From this we obtain the first explicit eww (exhaustive weakly wandering) sequence for the transformation. The exhaustive property of th...

The appearance of weakly wandering (ww) sets and sequences for ergodic transformations over half a century ago was an unexpected and surprising event. In time it was shown that ww and related sequences reflected significant and deep properties of ergodic transformations that preserve an infinite measure. This monograph studies in a systematic w...

About fifty years ago, questions on the existence and non-existence of finite invariant measures were studied by various authors and from different directions. In this article, we examine these classical results and prove directly that all the conditions introduced by these authors are equivalent to each other. We begin at the fundamental level of...

This book provides a self-contained introduction to typical properties of homeomorphisms. Examples of properties of homeomorphisms considered include transitivity, chaos and ergodicity. A key idea here is the interrelation between typical properties of volume preserving homeomorphisms and typical properties of volume preserving bijections of the un...

We dedicate this paper to the memory of Shizuo Kakutani. We miss his kind manner, gentle presence and keen insight. Abstract. We show that three theorems about the measurable dynamics of a fixed aperiodic measure preserving transformation τ of a Lebesgue probability space (X, A, µ) are equivalent. One theorem asserts that the conjugates of τ is den...

Wang tiles are square unit tiles with colored edges. A finite set of Wang tiles is a valid tile set if the collection tiles the plane (using an unlim- ited number of copies of each tile), the only requirements being that adjacent tiles must have common edges with matching colors and each tile can be put in place only by translation. In 1995 Kari an...

Suppose G is an infinite Abelian group that factorizes as the direct sum G = A ⊕ B: i.e., the B-translates of the single tile A, evenly tile the group G (B is called the tile set). In this note we consider conditions for another set C ⊂ G to tile G with the same tile set B. In an earlier paper we answered a question of Sands regarding such tilings...

We call an ordered set c = (c(i) : i ∈ N), of nonnegative extended real numbers c(i), a universal skyscraper template if it is the distribution of first return times for every ergodic measure preserving transformation T of an infinite Lebesgue measure space. If P i c(i) < ∞, we give a family of examples of ergodic infinite measure preserving transf...

Joel E. Cohen (1981) conjectured that any stochastic matrix P = fp i;j g could be represented by some circle rotation f in the following sense: For some par-tition fS i g of the circle into sets consisting of …nite unions of arcs, we have (*) p i;j = (f (S i) \ S j) == (S i), where denotes arc length. In this paper we show how cycle decomposition t...

We dedicate this paper to Shizuo Kakutani (1911–2004). Abstract. We consider three theorems in ergodic theory concerning a fixed aperiodic measure preserving transformation σ of a Lebesgue probability space (X, A, µ) and show that each theorem is a corollary of any other. One theorem asserts that the conjugates of σ are dense in the space of automo...

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair (a,b) of positive odd integers good, if Z=aS⊖2bS, where S is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair (a,b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is...

Using elementary methods, a positive answer is given to a question of A. D. Sands concerning fac-torizations of abelian groups. We then indicate how our approach to Sands's question has its roots in a result on the ergodic theory of infinite measure preserving transformations due to Eigen, Ha-jian and Ito.

First we describe the work of the first author leading to the conclusion that any property generic (in the weak topology) for measure-preserving bijections of a Lebesgue probability space is also generic (in the compact-open topology) for homeomorphisms of a compact manifold preserving a fixed measure. Then we describe the work of both authors in e...

While they were Junior Fellows at Harvard in the 1930's, J. C. Oxtoby and S. M. Ulam worked on the conjecture of G. D. Birkhoff and E. Hopf that ergodicity is the `general case' for homeomorphisms of a manifold preserving a fixed measure. Under the guidance of M. Stone, they proved this conjecture in an important paper published in 1941. Much later...

Let μ be a locally positive Borel measure on a σ-compact n-manifold X,n≥2. We show that there is always a μ-preserving homeomorphism of X which is maximally chaotic in that it satisfies Devaney's definition of chaos, with the sensitivity constant chosen maximally. Furthermore, maximally chaotic homeomorphisms are compact-open topology dense in the...

This book provides a self-contained introduction to typical properties of homeomorphisms. Examples of properties of homeomorphisms considered include transitivity, chaos and ergodicity. A key idea here is the interrelation between typical properties of volume preserving homeomorphisms and typical properties of volume preserving bijections of the un...

The authors establish the existence of self-homeomorphisms of ℝ n , n≥2, which are chaotic in the sense of R. L. Devaney [An introduction to chaotic dynamical systems (1989; Zbl 0695.58002)], preserve volume and are spatially periodic. Moreover, they show that in the space of volume-preserving homeomorphisms of the n-torus with mean rotation zero,...

. S. Alpern has proved that an invertible antiperiodic measurable measure preserving transformation of a Lebesgue probability space can be represented by k towers of heights n1 ; : : : ; n k , with prescribed measures, provided that the heights have greatest common divisor 1. In this paper we give a simple proof of Alpern's theorem. It is elementar...

We say that a homeomorphism h of the base space X (which may be either the annulus or n-torus, n≥2) is rotationless if it is area-preserving and has a lift h∼ to the covering space X∼([0,1] × R or Rn) with mean translation zero (∫Ω(h∼(x)–x)dx=0, where Ω is [0,1] × [0,1]). We prove (Theorem 1) that in the space of rotationless homeomorphisms of X wi...

We give two combinatorial proofs of Franks′ Theorem, that an area preserving torus homeomorphism with mean rotation zero has a fixed point. The first proof uses Lax′s version of the Marriage Theorem. The second proof uses Euler′s Theorem on circuits in graphs and an explicit method of Alpern for decomposing a Markov chain into cycles. Our methods a...

Let k0(Mn,π.) denote the set of all homeomorphisms of a compact manifold M" that preserve a locally positive nonatomic Borel probability measure p. and are isotopic to the identity. The notion of the mean rotation vector for a torus homeomorphism has been extended by Fathi to a continuous map θ on k0(Mn,π.). We show that any abstract ergodic behavi...

This paper is about typical (uniform topology dense Gδ) properties of homeomorphisms of the torus or annulus which preserve a fixed measure and have mean rotation zero. We first show that ergodicity is typical (Theorem 1). We then show that the lift (to the universal covering space) of such a homeomorphism of the annulus is the skew product of the...

Let σ be a nonsingular antiperiodic automorphism of a Lebesgue probability space . Let π = (π1, π2, …, πk, …) be a probability vector with the property that the k's for which πk> 0 is a relatively prime set of integers and ∑i = 1∞iπi < ∞. Then there is a measurable set B of positive measure such that the relative distribution of return times under...

In this paper we consider the problem of coding a given stationary stochastic process to another with a prescribed marginal distribution. This problem after reformulation is solved by proving the following theorem. Let $(M, \mathscr{A}, \mu)$ be a Lebesgue probability space and let $\sigma$ be an antiperiodic bimeasurable $\mu$-preserving automorph...

In this paper we consider a question concerning the conjugacy class of an arbitrary ergodic automorphism σ of a sigma finite Lebesgue space ( X , , μ ) (i.e., a is a ju-preserving bimeasurable bijection of ( X , , μ ). Specifically we prove THEOREM 1. Let τ, σ be any pair of ergodic automorphisms of an infinite sigma finite Lebesgue space ( X , , μ...

Let denote the group of homeomorphisms of a σ-compact manifold M which preserve a locally positive non-atomic measure μ. Such a manifold can be compactified by adjoining ideal points called ‘ends’, collectively denoted by E. Every homeomorphism h in induces a measure preserving system (E, , μ*, h*) where is the algebra of clopen subsets of E, μ* is...

Let denote the group of all homeomorphisms of a σ-compact manifold which preserve a σ-finite, nonatomic, locally positive and locally finite measure μ . In two recent papers [ 4, 5 ] the possible ergodicity of a homeomorphism h in was shown to be related to the homeomorphism h * induced by h on the ends of M . An end of a manifold is, roughly speak...

The recent paper of Berlanga and Epstein [5] demonstrated the significant role played by the “ends” of a noncompact manifold M in answering questions relating homeomorphisms of M to measures on M. In this paper we show that an analysis of the end behaviour of measure preserving homeomorphisms of a manifold also leads to an understanding of some of...

The authors prove that in the space of nonsingular transformations of a Lebesgue probability space the type III1 ergodic transformations form a denseG set with respect to the coarse topology. They also prove that for any locally compact second countable abelian groupH, and any ergodic type III transformationT, it is generic in the space ofH-valued...

We show that every compact connected Hubert cube manifold M can be obtained from the Hubert cube Q by making identifications on a face of Q. Some applications of this result to measure preserving homeomorphisms on M are given: (1) The first is concerned with which measures on M are equivalent to each other by homeomorphisms. (2) The second applicat...

Sans rsum

Let G be a locally compact abelian group. Then there is a finitely additive regular set function m defined on an algebra A of Borel sets in G, m(G) = 1, such that m(T-1 F) = m(F) for all F A and all surjective group endomorphisms T of G onto G.

Except for a set of first category, all pairs of measure preserving transformations generate a dense subgroup of $G$, the group of all invertible measure preserving transformations of the unit interval when $G$ has the weak topology.

Except for a set of first category, all pairs of measure preserving transformations generate a dense subgroup of G, the group of all invertible measure preserving transformations of the unit interval when G has the weak topology.

A Borel probability measure μ in the Hubert cube is homeomorphic to the usual product measure if and only if it is positive for nonempty open sets and zero for points. The transformation can be effected by a homeomorphism equal to the identity on any prescribed μ-null Z-set. Several extension, approximation, and embedding theorems are obtained as a...

Furstenberg conjectured that any continuous probability measure on (0; 1) invariant under multiplication by two and multiplication by three (denoted by R2(x) = 2x mod 1 and R3(x) = 3x mod 1) must be Lebesgue measure. Lyons showed this under the additional assumption that is exact for at least one of the transformations. Rudolph established Furstenb...

Written for the Dept. of Mathematics. Thesis (M.Sc.). Bibliography: leaves 72.