Ethan Akin

Li–Yorke sensitivity

We introduce and study a concept which links the Li–Yorke versions of chaos with the notion of sensitivity to initial conditions. We say that a dynamical system (X,T) is Li–Yorke sensitive if there exists a positive e such that every xX is a limit of points yX such that the pair (x,y) is proximal but not e-asymptotic, i.e. for infinitely many positive integers i the distance ρ(Ti(x),Ti(y)) is greater than e but for any positive δ this distance is less than δ for infinitely many i.

Dynamics of games and genes: Discrete versus continuous time

It is shown that in the classical model of population genetics (Fisher-Wright-Haldane, discrete or continuous version) every solution p(t) converges to equilibrium for t → ∞. For related models of evolutionary games (with non-symmetric matrices) it is shown that the transformation that describes the dynamics is a diffeomorphism (in particular one-to-one).

Residual properties and almost equicontinuity

A propertyP of a compact dynamical system (X,f) is called a residual property if it is inherited by factors, almost one-to-one lifts and surjective inverse limits. Many transitivity properties are residual. Weak disjointness from all propertyP systems is a residual property providedP is a residual property stronger than transitivity. Here two systems are weakly disjoint when their product is transitive. Our main result says that for an almost equicontinuous system (X, f) with associated monothetic group Λ, (X, f) is weakly disjoint from allP systems iff the onlyP systems upon which Λ acts are trivial. We use this to prove that every monothetic group has an action which is weak mixing and topologically ergodic.

Cycling in simple genetic systems

We describe the — unexpected — occurrence of stable limit cycles in the two locus, two allele model. No frequency dependence is involved. The cycles are due to the interaction between recombination and natural selection.

Evolutionary Dynamics of zero-sum games

Aim model in terms of differential equations is used to explain mammalian ovulation control, in particular regulation for a prescribed number of mature eggs.

Recurrence of the unfit

Abstract Domination among strategies in an evolutionary game implies that the geometric mean of the frequencies of certain strategies—the unfit—approaches zero. However, as we show by example no one strategy need be eliminated in the limit.

MANIFOLD PHENOMENA IN THE THEORY OF POLYHEDRA

Introduction. When can an isotopy be covered by an ambient isotopy? Let us restrict attention to the compact p.l. category. Hudson and Zeeman have shown that a locally unknotted isotopy of a manifold in a manifold can be covered by an ambient isotopy of the big manifold. By Zeemans codimension > 3 unknotting theorem, an isotopy of manifolds is locally unknotted if the codimension is greater than or equal to 3. Hence, any isotopy of a manifold in a manifold of dimension at least 3 higher can be covered. Lickorish has generalized Zeemans unknotting theorem to the case of a proper embedding of a cone in a ball of dimension at least three higher. From this, Hudson has shown any isotopy of a polyhedron in a manifold can be covered if the polyhedron has codimension at least 3. As a modest aim we would like a criterion of local unknottedness of a polyhedron in a manifold so that the original Hudson-Zeeman theorem would generalize. What we actually obtain is more general. We present a characterization of those isotopies of a polyhedron in a polyhedron which can be covered by ambient isotopies. Perhaps surprisingly, this question admits a rather elegant general solution. Intrinsic Dimension. Our major tool is the theory of intrinsic dimension developed by Armstrong. Given a polyhedral pair (X, X0) and a point x E X0, we define the intrinsic dimension of x in (X, X0), denoted d(x; X, X0), to be

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.

The Differential Geometry of Population Genetics and Evolutionary Games

Via the Riemannian metric introduced by Shahshahani and Conley, some elementary ideas from differential geometry have proved broadly useful in mathematical biology. The mathematics is not very deep but it is unfamiliar to many, so we begin with a survey of the elements of linear algebra on Euclidean vector spaces and of calculus on Riemannian manifolds. We then apply the Shahshahani metric to population genetics, deriving the occurrence of cycling in the two locus, two allele model. Then we turn to evolutionary games and compare the ESS condition with other notions of stability.

The topological dynamics of Ellis actions

Introduction Semigroups, monoids and their actions Ellis semigroups and Ellis actions Continuity conditions Applications using ideals Classical dynamical systems Classical actions: The group case Classical actions: The Abelian case Iterations of continuous maps Table Bibliography Index