The Iterated Prisoner's Dilemma: Good Strategies and Their Dynamics

For the iterated Prisoner's Dilemma, there exist Markov strategies which solve the problem when we restrict attention to the long term average payoff. When used by both players these assure the cooperative payoff for each of them. Neither player can benefit by moving unilaterally any other strategy, i.e. these are Nash equilibria. In addition, if a player uses instead an alternative which decreases the opponent's payoff below the cooperative level, then his own payoff is decreased as well. Thus, if we limit attention to the long term payoff, these \emph{good strategies} effectively stabilize cooperative behavior. We characterize these good strategies and analyze their role in evolutionary dynamics

Maximal r-Diameter Sets and Solids of Constant Width

We recall the definition of an r-maximal set in a metric space as a maximal subset of diameter r. In the special case when the metric space is Euclidean such a set is exactly a solid of constant diameter r. In the process of reviewing the theory of these objects we provide a simple construction which generates a large class of such solids

WAP Systems and Labeled Subshifts

The main object of this work is to present a powerful method of construction of subshifts which we use chiefly to construct WAP systems with various properties. Among many other applications of this so called labeled subshifts, we obtain examples of null as well as non-null WAP subshifts, WAP subshifts of arbitrary countable (Birkhoff) height, and completely scrambled WAP systems of arbitrary countable height. We also construct LE but not HAE subshifts, and recurrent non-tame subshifts.Comment: A revised versio

Chain transitive homeomorphisms on a space: all or none

We consider which spaces can be realized as the omega limit set of the discrete time dynamical system. This is equivalent to asking which spaces admit a chain transitive homeomorphism and which do not. This leads us to ask for spaces where all homeomorphisms are chain transitive

Generalized Intransitive Dice II: Partition Constructions

A generalized $N$-sided die is a random variable $D$ on a sample space of $N$ equally likely outcomes taking values in the set of positive integers. We say of independent $N$ sided dice $D_i, D_j$ that $D_i$ beats $D_j$, written $D_i \to D_j$, if $Prob(D_i > D_j) > \frac{1}{2}$. A collection of dice $\{ D_i : i = 1, \dots, n \}$ models a tournament on the set $[n] = \{ 1, 2, \dots, n \}$, i.e. a complete digraph with $n$ vertices, when $D_i \to D_j$ if and only if $i \to j$ in the tournament. By using $n$-fold partitions of the set $[Nn]$ with each set of size $N$ we can model an arbitrary tournament on $[n]$. A bound on the required size of $N$ is obtained by examples with $N = 3^{n-2}$

Chain Recurrence For General Spaces

The chain relation, due to Conley, and the strong chain relation, due to Easton, are well studied for continuous maps on compact metric spaces. Following Fathi and Pageault, we use barrier functions to generalize the theory to general relations on uniform spaces. In developing the theory, we indicate why the chain ideas are naturally uniform spaces concepts. We illustrate that the extension to relations is easy and is useful even for the study of the continuous map case

Compactifications of Dynamical Systems

While compactness is an essential assumption for many results in dynamical systems theory, for many applications the state space is only locally compact. Here we provide a general theory for compactifying such systems, i.e. embedding them as invariant open subsets of compact systems. In the process we don't want to introduce recurrence which was not there in the original system. For example if a point lies on an orbit which remains in any compact set for only a finite span of time then the point becomes non-wandering if we use the one-point compactification. Instead, we develop here the appropriate theory of dynamic compactification

Is there a Ramsey-Hindman theorem ?

We show that there does not exist a joint generalization of the theorems of Ramsey and Hindman, or more explicitly, that the property of containing a symmetric IP-set is not divisible

Varieties of Mixing

We consider extensions of the notion of topological transitivity for a dynamical system $(X,f)$. In addition to chain transitivity, we define strong chain transitivity and vague transitivity. Associated with each there is a notion of mixing, defined by transitivity of the product system $(X \times X, f \times f)$. These extend the concept of weak mixing which is associated with topological transitivity. Using the barrier functions of Fathi and Pageault, we obtain for each of these extended notions a dichotomy result that a transitive system of each type either satisfies the corresponding mixing condition or else factors onto an appropriate type of equicontinuous minimal system. The classical dichotomy result for minimal systems follows when it is shown that a minimal system is weak mixing if and only if it is vague mixing

Generically there is but one self homeomorphism of the Cantor set

We describe a self-homeomorphism $R$ of the Cantor set $X$ and then show that its conjugacy class in the Polish group $H(X)$ of all homeomorphisms of $X$ forms a dense $G_\delta$ subset of $H(X)$. We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class