Expansive self-homeomorphisms of the cantor set

Introduction. It was shown in [1] that the expansive homeomorphisms are dense in the group G of all self-homeomorphisms of the Cantor set when G has the pointwise topology. The purpose of this paper is to show that they are actually uniformly dense (i.e., dense in G with the compact-open topology) but that they form a set of first category. Because expansiveness is a property associated with the shift on the sequence space 1-I,~= _ ~ A,, where A, = {0, 2 } with the discrete topology, it is convenient to work in this space as well as the Cantor set. We call this space X(2) and the associated shift a. Let X denote the Cantor set in [0, 1]. Given an integer r > 1, we call [i3 -r , ( i+ 1)3-'] c~ X (i = 0, 1, 2 , . . , 3 r 1) a Cantor subinterval of rank r if ( i3' , ( i+ 1 ) 3 ' ) n X ¢ ~. Order the Cantor subintervals of rank r by the usual ordering of their left-hand end-points and denote the kth in this order by I(k, r) (k = 1, 2 , . , 2"). I f S e G, a cycle under S is a collection of Cantor subintervals of the same rank {I(k 1, r ) , . . . , I(ks, r)} such that SI(ki, r) = l(ki+l, r) for 1 < i < s and SI(ks, r) = I(kx, r).