Combinatorics of linear iterated function systems with overlaps

Let p₀, ..., p_m−1 be points in , and let be a one-parameter family of similitudes of : where λ ∊ (0, 1) is our parameter. Then, as is well known, there exists a unique self-similar attractor S_λ satisfying . Each x ∊ S_λ has at least one address , i.e. .

We show that for λ sufficiently close to 1, each x ∊ S_λ ∖ {p₀, ..., p_m−1} has different addresses. If λ is not too close to 1, then we can still have an overlap, but there exist xs which have a unique address. However, we prove that almost every x ∊ S_λ has addresses, provided S_λ contains no holes and at least one proper overlap. We apply these results to the case of expansions with deleted digits.

Furthermore, we give sharp sufficient conditions for the open set condition to fail and for the attractor to have no holes.

These results are generalizations of the corresponding one-dimensional results, however most proofs are different.