E. Arthur Robinson

Transformations with highly nonhomogeneous spectrum of finite multiplicity

This paper studies a spectral invariant ℳT for ergodic measure preserving transformationsT called theessential spectral multiplicities. It is defined as the essential range of the multiplicity function for the induced unitary operatorUT. Examples are constructed where ℳT is subject only to the following conditions: (i) 1∈ℳT, (ii) lcm(n, m)∈ℳT wherevern, m ∈ ℳT, and (iii) sup ℳT<+∞. This shows thatDT, definedDT=card ℳT, may be an arbitrary positive integer. The results are obtained by an algebraic construction together with approximation arguments.

A mathematical look at politics

Preface, for the Student Preface, for the Instructor Voting Two Candidates Scenario Two-candidate methods Supermajority and status quo Weighted voting and other methods Criteria Mays Theorem Exercises and problems Social Choice Functions Scenario Ballots Social choice functions Alternatives to plurality Some methods on the edge Exercises and problems Criteria for Social Choice Scenario Weakness and strength Some familiar criteria Some new criteria Exercises and problems Which Methods are Good? Scenario Methods and criteria Proofs and counterexamples Summarizing the results Exercises and problems Arrows Theorem Scenario The Condorcet paradox Statement of the result Decisiveness Proving the theorem Exercises and problems Variations on the Theme Scenario Inputs and outputs Vote-for-one ballots Approval ballots Mixed approval/preference ballots Cumulative voting . Condorcet methods Social ranking functions Preference ballots with ties Exercises and problems Notes on Part I Apportionment Hamiltons Method Scenario The apportionment problem Some basic notions A sensible approach The paradoxes Exercises and problems Divisor Methods Scenario Jeffersons method Critical divisors Assessing Jeffersons method Other divisor methods Rounding functions Exercises and problems Criteria and Impossibility Scenario Basic criteria Quota rules and the Alabama paradox Population monotonicity Relative population monotonicity The new states paradox Impossibility Exercises and problems The Method of Balinski and Young Scenario Tracking critical divisors Satisfying the quota rule Computing the Balinski-Young apportionment Exercises and problems Deciding Among Divisor Methods Scenario Why Webster is best Why Dean is best Why Hill is best Exercises and problems History of Apportionment in the United States Scenario The fight for representation Summary Exercises and problems Notes on Part II Conflict Strategies and Outcomes Scenario Zero-sum games The naive and prudent strategies Best response and saddle points Dominance Exercises and problems Chance and Expectation Scenario Probability theory All outcomes are not created equal Random variables and expected value Mixed strategies and their payouts Independent processes Expected payouts for mixed strategies Exercises and Problems Solving Zero-Sum Games Scenario The best response Prudent mixed strategies An application to counterterrorism The -by- case Exercises and problems Conflict and Cooperation Scenario Bimatrix games Guarantees, saddle points, and all that jazz Common interests Some famous games Exercises and Problems Nash Equilibria Scenario Mixed strategies The -by- case The proof of Nashs Theorem Exercises and Problems The Prisoners Dilemma Scenario Criteria and Impossibility Omnipresence of the Prisoners Dilemma Repeated play Irresolvability Exercises and problems Notes on Part III The Electoral College Weighted Voting Scenario Weighted voting methods Non-weighted voting methods Voting power Power of the states Exercises and problems Whose Advantage? Scenario Violations of criteria People power Interpretation Exercises and problems Notes on Part IV Solutions to Odd-Numbered Exercises and Problems Bibliography Index

Generalized -expansions, substitution tilings, and local finiteness

For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion β a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion β, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized β-transformation.

On the table and the chair

Abstract We find topological models for the tiling dynamical systems corresponding to the chair and table rep-tiles.

A general condition for lifting theorems

We define a general condition, called stability on extensions T of measure preserving transformation S. Stability is defined in terms of relative unique ergodicity, and as a joining property. Ergodic compact group extensions are stable, and moreover stable extensions satisfy lifting theorems similar to those satisfied by group extensions. In general, stable extensions have relative entropy zero. In the class of continuous flow extensions over strinctly ergodic homeomorphisms, stable extensions are generic

A weakly mixing tiling dynamical system with a smooth model

We describe a weakly mixing one-dimensional tiling dynamical system in which the tiling space is modeled by a surface M of genus 2. The tiling system satisfies an inflation, and the inflation map is modeled by a pseudo-Anosov diffeomorphism D on M. The expansion coefficient θ for D is a non-Pisot number. In particular, the leaves of the expanding foliation for D are tiled by their visits to the elements of a Markov partition for D. The tiling dynamical system is an almost 1:1 extension of the unit speed flow along these leaves.

Ergodic properties that lift to compact group extensions

Let T and R be measure preserving, T weakly mixing, R ergodic, and let S be conservative ergodic and nonsingular. Let T be a weakly mixing compact abelian group extension of T. If T x S is ergodic then T x S is ergodic. A corollary is a new proof that if T is mildly mixing then so is T. A similar statement holds for other ergodic multiplier properties. Now let T be a weakly mixing type a compact affine G extension of T where a is an automorphism of G. If T and R are disjoint and a or R has entropy zero, then T and R are disjoint. T is uniquely ergodic if and only if T is uniquely ergodic and as has entropy zero. If T is mildly mixing and T is weakly mixing then T is mildly mixing. We also provide a new proof that if T is weakly mixing then T has the K-property if T does. I. Statement of results. This paper is concerned with a general class of theorems called lifting theorems. A lifting theorem is a theorem of the following sort: Let T be a weakly mixing measure preserving transformation which satisfies an additional property P. If T is a weakly mixing extension of T, then T also satisfies P. Throughout this paper an extension will be a compact affine G extension of type (x, where G is a compact metrizable group and (x is a continuous automorphism of G (the definition is given in ?11). The case where (x is trivial is called a compact group extension (abelian extension if G is also abelian). Lifting theorems are useful for inductively constructing examples of transformations with various properties by lifting these properties to extensions. In the past, lifting theorems have been proven for various kinds of extensions for the following properties P: mixing, Thouvenot (unpublished); r-fold mixing, Rudolph [Rl]; the K-property, Parry [P] (cf. also [T]); and the Bernoulli property, Rudolph [R2]. In [Wi], Walters proved a general lifting theorem for abelian extensions concerning the absence of certain invariant sub-u-algebras. An easy corollary of this is a lifting theorem for mild mixing in abelian extensions (cf. Corollary 2.1). Berg [B] proved a lifting theorem for group extensions involving quasi-disjointness. In this paper we prove two new general lifting theorems, and discuss several corollaries. The proofs appear in ?11. We first consider measure spaces (X, ,u) and (Y, v), where X and Y are standard Borel spaces and ,u and v are nonatomic Borel probability measures. (We will Received by the editors January 24, 1986 and, in revised form, September 3, 1986. Some of the results in this paper were presented at the Conference on Smooth Ergodic Theory at the University of Warwick, Coventry, England, July 13, 1986. 1980 Mathematics Subject (Classijicatzon (1985 Ievi8sion). Primary 28D05, 28D20. Partially supported by NSF Grant DMS 85-04025. (?1988 American Mathematical Society 0002-9939/88

On the Existence of Markov Partitions for Zd Actions

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On Uniform Convergence in the Wiener-Wintner Theorem

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Non-abelian extensions have nonsimple spectrum

The theory of higher-dimensional shifts of finite type is still largely an open area of investigation. Recent years have seen much activity, but fundamental questions remain unanswered. In this paper we consider the following basic question. Given a shift of finite type (SFT), under what topological mixing conditions are we guaranteed the existence of Bernoulli (or even