On connected linearly ordered dynamical systems
by
Manuel Sanchis
Departament de Matemàtiques, Universitat Jaume I de Castelló (Spain)
Coauthors: Domingo Alcaraz (Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena (Spain))

The Sarkovski's ordering <= _S of N is defined by

1 <= S2 <= S22 <= S23 ... , ... <= S22.9 <= S22.7 <= S22.5 <= S22.3 <= S ... <= S2.9 <= S 2.7 <= S2.5 <= S2.3 <= S ... ... ... <= S9 <= S7 <= S5 <= S3,

A noteworthy theorem of Sarkovski characterizes the possible periods of orbits for a continuous function on the unit interval I:

(Sarkovski's Theorem) Let f in C(I ,  I). If f has a point of period m, then f has points of all periods preceding m in the Sarkovski's ordering. Conversely, if m is any positive integer, then there exists a continuous function f in C(I ,  I) having a point of period m, but having no points of period k for any k greater than m in the Sarkovski's ordering.

Further research starting at this point went in the direction of asking whether this theorem was valid in other spaces. For instance, it is well known that the same theorem is true if the unit interval I is replaced with the set of real numbers. In this framework, H. Schirmer (Houston J. Math., 1985) showed that the first part of Sarkovski's Theorem is valid for connected linearly ordered spaces and asked if the second part also works in this realm. Schirmer's question was answered by S. Baldwin (Houston J. Math., 1991) in an interesting and unexpected way. Baldwin's Theorem states that a connected linearly ordered spaces L satisfies Sarkovski's Theorem if and only if there exists a function in C(L,  L) having an orbit of period something other than a power of 2.

In this talk, starting from Bladwin's Theorem, several dynamical properties of linearly ordered dynamical systems are presented. For example, given a connected linearly ordered space L, we prove that for a continuous function f from L into itself, cl\sbLP(f)=cl\sbLR(f) where P(f) (respectively, R(f)) stands for the set of all periodic points (respectively, all recurrent points) of f. We also show that the following conditions are equivalent: (1) L satisfies Sarkovskii's Theorem, (2) there exist turbulent functions on L, and (3) there exists a compact subspace of L which satisfies Sarkovskii's Theorem. Our results are applied in two ways. Firstly, we show that there exist connected linearly ordered spaces without infinite minimal sets; secondly, for each uncountable cardinal number \lambda, we construct a connected linearly ordered space L such that: (1) L is a compact no first countable space satisfying Sarkovskii's Theorem, (2) L admits a dense first countable subset, and (3) the density of L is \lambda.

Date received: February 28, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-47.