arXiv subjects

Ethan Akin

Publications and source records attributed to Ethan Akin.

29 records · Page 2Linked to original sources

arXiv2017

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.

math.DS↗

arXiv2017

Good Strategies for the Iterated Prisoner's Dilemma : Smale vs. Markov

In 1980 Steven Smale introduced a class of strategies for the Iterated Prisoner's Dilemma which used as data the running average of the previous payoff pairs. This approach is quite different from the Markov chain approach, common before and since, which used as data the outcome of the just previous play, the memory-one strategies. Our purpose here is to compare these two approaches focusing upon good strategies which, when used by a player, assure that the only way an opponent can obtain at least the cooperative payoff is to behave so that both players receive the cooperative payoff. In addition, we prove a version for the Smale approach of the so-called Folk Theorem concerning the existence of Nash equilibria in repeated play. We also consider the dynamics when certain simple Smale strategies are played against one another.

math.DS↗

arXiv2016

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.

math.DS↗

arXiv2016

Variations on the Concept of Topological Transitivity

We describe various strengthenings of the concept of topological transitivity. Especially when one departs from the family of invertible systems, a number of interesting properties arise. We present the architecture of implications among ten reasonable notions of transitivity.

math.DS↗

arXiv2015

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.

math.DS↗

arXiv2014

Dynamics of Induced Systems

In this paper, we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if $X$ is a metric space, let $2^X$ denote the space of non-empty compact subsets of $X$ provided with the Hausdorff topology. If $f$ is a continuous self-map on $X$, there is a naturally induced continuous self-map $f_*$ on $2^X$. Our main theme is the interrelation between the dynamics of $f$ and $f_*$. For such a study, it is useful to consider the space $\mathcal{C}(K,X)$ of continuous maps from a Cantor set $K$ to $X$ provided with the topology of uniform convergence, and $f_*$ induced on $\mathcal{C}(K,X)$ by composition of maps. We mainly study the properties of transitive points of the induced system $(2^X,f_*)$ both topologically and dynamically, and give some examples. We also look into some more properties of the system $(2^X,f_*)$.

math.DS↗

arXiv2012

Dynamics of Discontinuous Maps Via Closed Relations

For the dynamics of a discontinuous map on a compact metric space, we describe an approach using suitable closed relations and connect it with the continuous dynamics on an invariant G-delta subset and with the continuous dynamics on the compact space of sample paths.

math.DS↗

arXiv2012

Conceptions of Topological Transitivity

There are several different common definitions of a property in topological dynamics called "topological transitivity," and it is part of the folklore of dynamical systems that under reasonable hypotheses, they are equivalent. Various equivalences are proved in different places, but the full story is difficult to find. This note provides a complete description of the relationships among the different properties.

math.DS↗

arXiv2010

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.

math.DS↗

arXiv2010

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.

math.DS↗

arXiv2006

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 $G_δ$ subset of $H(X)$. We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.

math.DS↗