Multiple codings of self-similar sets with overlaps

In this paper we consider a general class epsilon of self-similar sets with complete overlaps. Given a self-similar iterated function system Phi =( E, {f(i)}(i=1)(m)) is an element of epsilon o n the real line, for each point x is an element of E we can find a sequence (i(k)) = i(1)i(2) ... is an element of{1,..., m}(N), called a coding of x, such that x = lim(n ->infinity) fi(1) circle fi(2) circle ... circle fi(n) (0). For k= 1, 2,..., aleph(0) or 2(aleph 0) we investigate the subset U-k(Phi) which consists of all x is an element of E having precisely k different codings. Among several equivalent characterizations we show that U-1(Phi) is closed if and only if U-aleph 0 (Phi) is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of U-k(Phi), and show that the corresponding Hausdorff measure of U-k(Phi) is always infinite for any k >= 2. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in U-k(Phi) and U-aleph 0(Phi). (c) 2020 Elsevier Inc. All rights reserved.