On the periodic points of a typical continuous function

Abstract

Let n and k be arbitrary natural numbers. We prove that for a continuous f, every neighborhood of any periodic point of f with n contains periodic points of f with n * k . The purpose of this paper is to examine the structure of the set of periodic points of continuous functions. We shall denote by C the set of continuous functions mapping the interval [0, 1] into itself. This set becomes a metric space with the supremum metric. By the term typical continuous function we mean that the set of all functions having the property under consideration is a residual subset of the metric space C. In [1] S. J. Agronsky, A. M. Bruckner, and M. Laczkovich proved that for a continuous f any neighborhood of a periodic point contains periodic points of large periods. In this paper we prove that this result is true if arbitrarily large periods is replaced by period k * n (k = 1 , 2, ...) (Theorem 1). As a consequence, for a continuous f we have that Pf the set of periodic points with n, is uncountable, dense in itself, but not closed. Also, P) is a residual subset of Fix(fn). In [1] it is proved that, for a continuous f, Fix(f n) is nowhere dense and perfect. In this paper we prove (Theorem 2) that is bilaterally strongly (D-porous. For f E C and n E N we define fn (x) by f '(x) = f(x) and fn(x) = f(fn-l(X)). PJn denotes the set of periodic points of f with n. We denote = {x: fn(X) = x}; then P = Fix(fn) Uk 0 denote B(f ,e) = {g E C: Ilf gll < }. Theorem 1. Let n , k E N be arbitrary. Then for a continuous f, every neighborhood of any periodic point of f with n contains periodic points of f with n * k. Received by the editors August 3, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H20; Secondary 26A18. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page