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Ethan Akin

Publications and source records attributed to Ethan Akin.

At least 19 recordsLinked to original sources
arXiv2026

Realizing Arbitrary Depth

We provide a simple construction which realizes the Birkhoff center depth at an arbitrary ordinal level and relate it to the Cantor-Bendixson depth.

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arXiv2026

Invertibility, Often

By using a similar pattern of arguments, we show that in four categories the collection of isomorphisms forms a residual subset of the space of morphisms. We first consider surjective continuous mappings on Cantor spaces. Next, we look at measure preserving maps on Polish measure spaces. We then consider the $L^1$ representations of nonsingular maps on Polish measure spaces. Finally, we examine continuous, measure preserving maps on Cantor spaces equipped with so-called good measures.

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arXiv2024

The Simple Yield Curve Models

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, $- \stackrel{\bullet}{P}_t/P_t$ defines the yield curve function $Y_t$. An $n$ parameter linear yield curve model selects as allowable yield curves $Y_t(r) = \sum_{i=1}^n r_i Y^i_t$ with the functions $Y^i$ fixed and with $r$ varying over an open subset of $\mathbb{R}^n$ on which $Y_t(r) \ge 0$ for all $t \in \mathbb{R}_+$. For example, the flat yield curve model with $P_t(r) = e^{-rt}$ is a one parameter linear model with $Y^1_t(r) = r > 0$. We impose two natural economic requirements on the model: (SPA) static prices allowed, i.e. it is always possible that as time moves forward, relative prices do not change, and (NLA) no local arbitrage, i.e. there does not exist a self-financing bundle of futures such that the zero present value is a local minimum with respect to small changes in the space of admissible yield curves. In that case the model always contains one of four simple models. If we impose the additional requirement (LRE) long rates exist, i.e. for every $r$ $Lim_{t \to \infty} Y_t(r)$ exists as a finite limit, then the number of simple models is reduced to two.

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arXiv2023

Dynamical Systems: Discrete, Continuous and Hybrid

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 relation extension) as well as hybrid dynamical systems which combine both continuous time and discrete time dynamics. In a unified way we describe the attractor-repeller structure, Conley's chain recurrence relation and the construction of Lyapunov functions for all of these systems.

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arXiv2023

Topological Tournaments

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 describe a tournament $R \subset X \times X$ as a total, antisymmetric relation, i.e. $R \cup R^{-1} = X \times X$ and $R \cap R^{-1}$ is the diagonal $1_X = \{ (x,x) : x \in X \}$. The set of arcs is $R^{\circ} = R \setminus 1_X = (X \times X) \setminus R^{-1}$. A topological tournament on a compact Hausdorff space $X$ is a tournament $R$ which is a closed subset of $X \times X$. We construct uncountably many non-isomorphic examples on the Cantor set $X$ as well as examples of arbitrarily large cardinality. We also describe compact Hausdorff spaces which do not admit any topological tournament.

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arXiv2021

Trees and Homogeneous LOTS

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 distinct CHLOTS. Using tree constructions we are able to extend the towers and to describe an inductive procedure which yields every CHLOTS.

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arXiv2020

Rock, Paper, Scissors, Etc -- Topics in the Theory of Regular Tournaments

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 where each strategy beats and is beaten by exactly half of the alternatives. We survey the properties of regular tournaments, which we will call games. In the process we describe a number of constructions for such games. These include games on groups of odd order and the associated games on coset spaces. We obtain a new lower bound for the number of games of size 2n+1.

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arXiv2019

Conjugacy in the Cantor Set Automorphism Group

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 defined subsets of $C_s(X)$, e. g. the sets of topologically transitive maps, chain transitive maps, chain mixing maps, etc. In each case, we consider whether there exist elements with a dense conjugacy class and if so, what the generic elements look like.

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arXiv2019

Approximation Dynamics

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 measure, almost every point is generic for one of a finite number of ergodic invariant measures. The approximations use non-degenerate simplicial dynamical systems for p. l. manifolds and shift-like dynamical systems for Cantor Sets.

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arXiv2019

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 $N = 3^{n-2}$.

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arXiv2019

Generalized Intransitive Dice: Mimicking an Arbitrary Tournament

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} $. Examples are known of intransitive $6$-sided dice, i.e. $D_1 \to D_2 \to D_3$ but $D_3 \to D_1$. A tournament of size $n$ is a choice of direction $i \to j$ for each edge of the complete graph on $n$ vertices. We show that if $R$ is tournament on the set $\{ 1, \dots, n \}$, then for sufficiently large $N$ there exist sets of independent $N$-sided dice $\{ D_1, \dots, D_n \}$ such that $D_i \to D_j$ if and only if $i \to j$ in $R$.

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arXiv2018

On weak rigidity and weakly mixing enveloping semigroups

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 introductory section recalling some definitions and classic results, we establish some necessary conditions for this to happen, and in the final section we show, using Ratner's theory, that the enveloping semigroup of the `time one map' of a classical horocycle flow is weakly mixing.

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arXiv2018

Induced Fuzzy Topological Spaces: A Characterization

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 compactness for such spaces.

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arXiv2017

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.

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arXiv2017

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.

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