Intersections of homogeneous Cantor sets and beta-expansions

Abstract

Let Γ β,N be the N -part homogeneous Cantor set with β ∈ (1/(2N − 1), 1/N ).is called a code of t. Let U β,±N be the set of t ∈ [−1, 1] having a unique code, and let S β,±N be the set of t ∈ U β,±N which make the intersection Γ β,N ∩ (Γ β,N + t) a selfsimilar set. We characterize the set U β,±N in a geometrical and algebraical way, and give a sufficient and necessary condition for t ∈ S β,±N . Using techniques from beta-expansions, we show that there is a critical point βc ∈ (1/(2N − 1), 1/N ), which is a transcendental number, such that U β,±N has positive Hausdorff dimension if β ∈ (1/ (2N − 1), βc), and contains countably infinite many elements if β ∈ (βc, 1/N ). Moreover, there exists a second critical point αc = N + 1 − (N − 1)(N + 3) /2 ∈ (1/ (2N − 1), βc) such that S β,±N has positive Hausdorff dimension if β ∈ (1/ (2N − 1), αc), and contains countably infinite many elements if β ∈ [αc, 1/N ).

On the continuity of the Hausdorff dimension of the univoque set

“…The number q KL is called the Komornik-Loreti constant; see Section 2 below for a precise definition. The above result was generalized to arbitrary M ≥ 1 by Baker [7] and Kong et al [22].…”

Entropy, topological transitivity, and dimensional properties of unique 𝑞-expansions

“…[16], see also, [22]). One might expect that for each connected component (q 0 , q * 0 ) of (q c , M + 1] \ U , the closed interval [q 0 , q * 0 ] would be an entropy plateau of H. However this is not true.…”

An algebraic approach to entropy plateaus in non-integer base expansions

“…Later, Glendinning and Sidorov [12] showed that U q is countably infinite for q G < q < q KL ; uncountable but of zero Hausdorff dimension for q = q KL ; and of positive Hausdorff dimension for q > q KL . These results were generalized to larger alphabets by Baker [6] and Kong, Li and Dekking [17]. Kong and Li [16] further examined the Hausdorff dimension of U q , and more recently, Komornik, Kong and Li [13] showed that this dimension is related to the topological entropy of the symbolic univoque set…”