On Diffeomorphisms of Compact 2-Manifolds with All Nonwandering Points Being Periodic

The aim of the present paper is to study conditions under which all the nonwandering points are periodic points, for a discrete dynamical system of two variables defined on a compact manifold. We include a survey of known results in all dimensions, and study the remaining open question in dimension two. We present two results, one negative and one positive. The negative result: we construct a Kupka–Smale diffeomorphism in [Formula: see text] (which can be extended to a diffeomorphism of the sphere) with a closed set of periodic points that differs from the set of nonwandering points. The positive result: we present a condition on the widely studied Hénon family which guarantees that all nonwandering points are periodic. Finally, we close by describing what future work may be needed to resolve our broad goals.

ON THE DYNAMICAL BEHAVIOR OF CELLULAR AUTOMATA

In this paper, we show that the mathematical classification of one-dimensional cellular automata given by Dubois–Violette and Rouet [1987] does not truly represent the different behaviors possible. In fact we show that many rules of their type O have totally different behavior. A better classification is based on Smale's basic sets and nonwandering points.

Nonwandering sets of maps on the circle

Letfbe a continuous map of the circleS1into itself. And letR(f),Λ(f),Γ(f), andΩ(f)denote the set of recurrent points,ω-limit points,γ-limit points, and nonwandering points off, respectively. In this paper, we show that each point ofΩ(f)\R(f)¯is one-side isolated, and prove that(1)Ω(f)\Γ(f)is countable and(2)Λ(f)\Γ(f)andR(f)¯\Γ(f)are either empty or countably infinite.

A CHARACTERIZATION OF THE SET Ω (f)\ ω (f) FOR CONTINUOUS MAPS OF THE INTERVAL WITH ZERO TOPOLOGICAL ENTROPY

We give a characterization of the set of nonwandering points of a continuous map f of the interval with zero topological entropy, attracted to a single (infinite) minimal set Q. We show that such a map f can have a unique infinite minimal set Q and an infinite set B ⊂ Ω (f)\ ω (f) (of nonwandering points that are not ω-limit points) attracted to Q and such that B has infinite intersections with infinitely many disjoint orbits of f.

Jacobi matrices and transversality

SynopsisThe paper deals with smooth nonlinear ODE systems in ℝn, ẋ = f(x), such that the derivative f′(x) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f′(x(t))y. Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝn is then explicitly constructed.