Richard Swanson

Invariants of weak equivalence in primitive matrices

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one dimensional inverse limit spaces.

Pseudo-orbits and topological entropy

The topological entropy of a map of a compact metric space is equal to the exponential growth rate of the number of separated periodic pseudo-orbits of period n as n tends to infinity.

Genericity of the fixed point set for the infinite population genetic algorithm

The infinite populationmodel for the genetic algorithm,where the iteration of the genetic algorithm corresponds to an iteration of a map G, is a discrete dynamical system. The map G is a composition of a selection operator and a mixing operator, where the latter models the effects of both mutation and crossover. This paper shows that for a typical mixing operator, the fixed point set of G is finite. That is, an arbitrarily small perturbation of the mixing operator will result in a map G with finitely many fixed points. Further, any sufficiently small perturbation of the mixing operator preserves the finiteness of the fixed point set.

Topological conjugacy of singularities of vector fields

ROUGHLY speaking, a smooth vector field is finitely determined for topological conjugacy at a singularity if its conjugacy class contains some finite Taylor approximation. Finite determinacy is one criterion for the existence of conjugacies between smooth planar vector fields. This criterion can fail at a singularity only if the vector field is either P-flat or not sufficiently smooth. In both situations we construct specific examples of singularities. not rotation points, that are topologically equivalent but not conjugate. In addition, we determine a useful necessary and sufficient condition for the conjugacy of a large class of singularities. This allows for a simple characterization of the resulting conjugacy classes.

Asymptotic entropy, periodic orbits, and pseudo-Anosov maps

In this paper we derive some properties of a variety of entropy that measures rotational complexity of annulus homeomorphisms, called asymptotic or rotational entropy. In previous work [ KS ] the authors showed that positive asymptotic entropy implies the existence of infinitely many periodic orbits corresponding to an interval of rotation numbers. In our main result, we show that a Holder

Boundaries of rotation sets for homeomorphisms of the -torus

C^1

Immediate conditional hyperbolicity in dynamical systems

diffeomorphism with nonvanishing asymptotic entropy is isotopic rel a finite set to a pseudo-Anosov map. We also prove that the closure of the set of recurrent points supports positive asymptotic entropy for a (

Nonconjugate pointed generalized solenoids with shift equivalent ₁-actions

C^0

Topological hyperbolicity and the Coleman Conjecture for diffeomorphisms

) homeomorphism with nonzero asymptotic entropy.

Minimal sets of periods for torus maps via Nielsen numbers

We construct a Cω diffeomorphism of the 3-torus whose rotation set is not closed. We prove that the rotation set of a homeomorphism of the n-torus contains the extreme points of its closed convex hull. Finally, we show that each pseudo-rotation set is closed for torus homeomorphisms.