A note on unstable homeomorphisms

Abstract

In [1 ] W. R. Utz introduced the concept of an unstable2 homeomorphism and raised the question of whether there exists an unstable homeomorphism of a compact continuum onto itself. In this note an example of such an homeomorphism will be given. Let C denote the complex unit circle and for each zE C, let g(z) =Z2. Then g: C onto C determines an inverse limit space 12 = {(ao, al, a2, * * ) J for each non-negative integer i, aiE C and g(ai+1) =ai}. For a, bC22, the function p(a, b) = , 1o Jai-bi /2i is a metric for 12; 12 is familiar as the two-solenoid, and is a compact, indecomposable continuum. Define f:22 onto 12 as follows: for each a=(ao, al, EZ*)2, let f(a) = [g(ao), g(al), * . Then f(a) =(a2, a2, *. ) =(aao, ao1 al, . ),f(a) = (a,, a2, a3, ),andf is a homeomorphism of 12 onto 22. To show that f is unstable, suppose that a= (ao, al, ) and b = (bo, b1, * * * ) are distinct points of 22. Consider, as Case 1, that ao#bo. Let ei?=ao, eiO=bo, where 0 1 Case 2: for some integer n> O, an F$bn, but ai = bi, for 0 i <n. Then f-n(a) = (ans, an+l, afn+2, * * * ), f-n(b) = (bn, bn+l, bn+2, * * * ), and there-