Ethan Akin
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View article: Conditional cooperation with longer memory
Conditional cooperation with longer memory Open
Direct reciprocity is a wide-spread mechanism for the evolution of cooperation. In repeated interactions, players can condition their behavior on previous outcomes. A well-known approach is given by reactive strategies, which respond to th…
View article: A Complete Invariant for Shift Equivalence for Boolean Matrices and Finite Relations
A Complete Invariant for Shift Equivalence for Boolean Matrices and Finite Relations Open
We give a complete invariant for shift equivalence for Boolean matrices (equivalently finite relations), in terms of the period, the induced partial order on recurrent components, and the cohomology class of the relation on those component…
View article: The Simple Yield Curve Models
The Simple Yield Curve Models Open
With $P_t$ the price in current dollars of a dollar delivered $t$ time units from now, we assume that $P$ is a decreasing function defined for $t \in \mathbb{R}_+$ with $P_0 = 1$. The negative logarithmic derivative, $- \stackrel{\bullet}{…
View article: Dynamical Systems: Discrete, Continuous and Hybrid
Dynamical Systems: Discrete, Continuous and Hybrid Open
The dynamics by iteration of a function on a compact metric space, sometimes called a cascade, can be extended to the dynamics of a closed relation on such a space. Here we apply this relation dynamics to study semiflows (and their relatio…
View article: Topological Tournaments
Topological Tournaments Open
A directed graph $R^{\circ}$ on a set $X$ is a set of ordered pairs of distinct points called \emph{arcs}. It is a tournament when every pair of distinct points is connected by an arc in one direction or the other (and not both). We can de…
View article: Trees and Homogeneous LOTS
Trees and Homogeneous LOTS Open
We describe those complete linearly ordered topological spaces $X$ which are homogeneous (=CHLOTS). That is, $X$ is order isomorphic with any nonempty open interval in $X$. Using countable tail-like ordinals as indices, we build towers of …
View article: Generalized intransitive dice II: Partition constructions
Generalized intransitive dice II: Partition constructions Open
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…
View article: Generalized intransitive dice: Mimicking an arbitrary tournament
Generalized intransitive dice: Mimicking an arbitrary tournament Open
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…
View article: Asymptotic Pairs for Interval Exchange Transformations
Asymptotic Pairs for Interval Exchange Transformations Open
We provide a simple description of asymptotic pairs in the subshift associated with an interval exchange transformation and show that, under reasonably general conditions, doubly asymptotic pairs do not occur.
View article: On weak rigidity and weakly mixing enveloping semigroups
On weak rigidity and weakly mixing enveloping semigroups Open
The question we deal with here, which was presented to us by Joe Auslander and Anima Nagar, is whether there is a nontrivial cascade (X,T) whose enveloping semigroup, as a dynamical system, is topologically weakly mixing (WM). After an int…
View article: Conjugacy in the Cantor Set Automorphism Group
Conjugacy in the Cantor Set Automorphism Group Open
We survey, and extend, results on the adjoint action of the homeomorphism group $H(X)$ on the space of surjective continuous maps, $C_s(X)$, where $X$ is a Cantor set. We look also at the restriction of the action to various dynamically de…
View article: Smale strategies for the n-person iterated prisoner's dilemma
Smale strategies for the n-person iterated prisoner's dilemma Open
Adapting methods introduced by Steven Smale, we describe good strategies for a symmetric version of the Iterated Prisoner's Dilemma with n players.
View article: Varieties of mixing
Varieties of mixing Open
We consider extensions of the notion of topological transitivity for a dynamical system . In addition to chain transitivity, we define strong chain transitivity and vague transitivity. Associated with each, there is a notion of mixing, def…
View article: Rock, Paper, Scissors, Etc -- Topics in the Theory of Regular Tournaments
Rock, Paper, Scissors, Etc -- Topics in the Theory of Regular Tournaments Open
The classic Rock-Paper-Scissors game of size 3 and its extension, Rock-Paper-Scissors-Lizard-Spock, are modeled by directed graphs called tournaments. They can be further extended to any odd size. The extended games are regular tournaments…
View article: Rock, Paper, Scissors, Etc - The Theory of Regular Tournaments
Rock, Paper, Scissors, Etc - The Theory of Regular Tournaments Open
Following the extension of the classic Rock-Paper-Scissors game of size 3 to Rock-Paper-Scissors-Lizard-Spock of size 5, we consider further extensions to larger odd numbers. Such extensions are modeled by directed graphs called tournament…
View article: Induced Fuzzy Topological Spaces: A Characterization
Induced Fuzzy Topological Spaces: A Characterization Open
We introduce a simple property, affine invariance, which characterizes within the class of fuzzy topological spaces those which are induced from an underlying topology on the space. We illustrate it by considering the simple notions of com…
View article: Chain transitive homeomorphisms on a space: all or none
Chain transitive homeomorphisms on a space: all or none Open
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…
View article: Chain Recurrence For General Spaces
Chain Recurrence For General Spaces Open
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 rela…
View article: Approximation Dynamics
Approximation Dynamics Open
We describe the approximation of a continuous dynamical system on a p. l. manifold or Cantor set by a tractable system. A system is tractable when it has a finite number of chain components and, with respect to a given full background meas…
View article: Good strategies for the Iterated Prisoner's Dilemma: Smale vs. Markov
Good strategies for the Iterated Prisoner's Dilemma: Smale vs. Markov Open
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 a…
View article: Good Strategies for the Iterated Prisoner's Dilemma : Smale vs. Markov
Good Strategies for the Iterated Prisoner's Dilemma : Smale vs. Markov Open
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 a…
View article: The iterated Prisoner’s Dilemma: good strategies and their dynamics
The iterated Prisoner’s Dilemma: good strategies and their dynamics Open
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 …
View article: Conjugacy in the Cantor set automorphism group
Conjugacy in the Cantor set automorphism group Open
We survey, and extend, results on the adjoint action of the homeomorphism group $H(X)$ on the space of surjective continuous maps, $C_s(X)$, where $X$ is a Cantor set. We look also at the restriction of the action to various dynamically de…
View article: Variations on the concept of topological transitivity
Variations on the concept of topological transitivity Open
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 t…
View article: What You Gotta Know to Play Good in the Iterated Prisoner’s Dilemma
What You Gotta Know to Play Good in the Iterated Prisoner’s Dilemma Open
For the iterated Prisoner’s Dilemma there exist good 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 pl…
View article: Is there a Ramsey-Hindman theorem ?
Is there a Ramsey-Hindman theorem ? Open
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.