Concerning periodic points in mappings of continua

Abstract

In this paper we present some conditions which are sufficient for a mapping to have periodic points. Theorem. If f f is a mapping of the space X X into X X and there exist subcontinua H H and K K of X X such that (1) every subcontinuum of K K has the fixed point property, (2) f [ K ] f[K] and every subcontinuum of f [ H ] f[H] are in class W W , (3) f [ K ] f[K] contains H H , (4) f [ H ] f[H] contains H ∪ K H \cup K , and (5) if n n is a positive integer such that ( f | H ) − n ( K ) {(f|H)^{ - n}}(K) intersects K K , then n = 2 n = 2 , then K K contains periodic points of f f of every period greater than 1. Also included is a fixed point lemma: Lemma. Suppose f f is a mapping of the space X X into X X and K K is a subcontinuum of X X such that f [ K ] f[K] contains K K . If (1) every subcontinuum of K K has the fixed point property, and (2) every subcontinuum of f [ K ] f[K] is in class W W , then there is a point x x of K K such that f ( x ) = x f(x) = x . Further we show that: If f f is a mapping of [ 0 , 1 ] [0,1] into [ 0 , 1 ] [0,1] and f f has a periodic point which is not a power of 2, then lim { [ 0 , 1 ] , f } \lim \{ [0,1],f\} contains an indecomposable continuum. Moreover, for each positive integer i i , there is a mapping of [ 0 , 1 ] [0,1] into [ 0 , 1 ] [0,1] with a periodic point of period 2 i {2^i} and having a hereditarily decomposable inverse limit.