Hausdorff dimension of double base expansions and binary shifts with a hole

1. Problem Statement and Context

The Core Problem:
The paper investigates the Hausdorff dimension of the univoque set $U_{q_0, q_1}$, which consists of real numbers having a unique $(q_0, q_1)$-expansion.

• Double Base Expansions: For two real bases $q_0, q_1 > 1$, a sequence $i_1 i_2 \dots \in \{0, 1\}^\infty$ represents a number $x$ via the series:
  $$x = \pi_{q_0, q_1}(i_1 i_2 \dots) = \sum_{k=1}^\infty \frac{i_k}{q_{i_1} q_{i_2} \dots q_{i_k}}$$
• The Univoque Set: $U_{q_0, q_1} = \{ x \in J_{q_0, q_1} : \#\pi_{q_0, q_1}^{-1}(x) = 1 \}$.
• The Gap: While the properties of univoque sets for a single base ($q_0 = q_1 = q$) are well-studied (e.g., by Komornik, Kong, Li, and Allaart), the case of distinct bases ($q_0 \neq q_1$) presents significant complexity. The authors aim to derive explicit formulas for the Hausdorff dimension of $U_{q_0, q_1}$ and prove the continuity of this dimension as a function of the pair $(q_0, q_1)$.

Dynamical Systems Connection:
The problem is framed within the context of open dynamical systems (dynamical systems with a "hole"). The set of unique expansions corresponds to the survivor set of a binary shift where sequences falling into a specific "hole" (defined by lexicographic bounds) are removed. This connects the problem to Lorenz maps and the doubling map with a hole.

2. Methodology

The authors employ a combination of symbolic dynamics, iterated function systems (IFS), and thermodynamic formalism.

• Symbolic Characterization:
  The set of unique expansions is characterized symbolically. Let $a_{q_0, q_1}$ be the quasi-greedy expansion of $1/q_1$ and $b_{q_0, q_1}$ be the quasi-lazy expansion of $1/(q_0(q_1-1))$. The set of unique expansions corresponds (up to a countable set) to the subshift:
  $$\Omega_{a, b} = \{ i_1 i_2 \dots \in \{0, 1\}^\infty : i_n i_{n+1} \dots \notin ]a, b[ \text{ for all } n \ge 1 \}$$
  where $]a, b[$ is the open interval in lexicographic order.

• Generalized Moran Construction:
  The authors analyze the projection $\pi_{q_0, q_1}(\Omega_{a, b})$. They treat this as a generalized Moran construction. The Hausdorff dimension is determined by the growth rate of the number of admissible words of length $n$, weighted by the contraction rates of the IFS maps $f_0(x) = x/q_0$ and $f_1(x) = (x+1)/q_1$.

• Kneading Invariants and Power Series:
  A crucial technical tool is the kneading invariant $K_{a,b}(z_0, z_1)$, defined via formal power series related to the boundaries $a$ and $b$. The authors establish a relationship between the topological entropy of the subshift and the roots of the equation:
  $$\pi_{\beta, \beta}(a) = \pi_{\beta, \beta}(b)$$
  Similarly, the Hausdorff dimension is linked to the root $s$ of:
  $$\pi_{q_0^s, q_1^s}(a) = \pi_{q_0^s, q_1^s}(b)$$

• Continuity Proofs:
  To prove continuity, the authors analyze the stability of the "hole" boundaries ($a$ and $b$) under small perturbations of the bases. They utilize substitution maps and show that the dimension function is locally constant or varies continuously by analyzing the behavior of the kneading invariants near critical points.

3. Key Contributions and Results

A. Explicit Formulas for Hausdorff Dimension

The paper provides a complete classification of the Hausdorff dimension $s = \dim_H U_{q_0, q_1}$ based on the relationship between $q_0$ and $q_1$:

1. Trivial/Countable Case: If $q_1 \le K(q_0)$ (where $K(q_0)$ is the generalized Komornik–Loreti constant), then $s = 0$.
2. Intermediate Case ($K(q_0) < q_1 < \frac{q_0}{q_0-1}$):
   The dimension $s$ is the maximal root of the equation:
   $$\pi_{q_0^s, q_1^s}(\ell) = \pi_{q_0^s, q_1^s}(r)$$
   where $\ell = \ell(a_{q_0, q_1}, b_{q_0, q_1})$ and $r = r(a_{q_0, q_1}, b_{q_0, q_1})$ are the extremal elements of the subshift $\Omega_{a, b}$.
   • Special Subcase: If the boundaries themselves are in the subshift, the equation simplifies to $\pi_{q_0^s, q_1^s}(a) = \pi_{q_0^s, q_1^s}(b)$.
3. Full Dimension Case: If $q_1 = \frac{q_0}{q_0-1}$, then $s = 1$.
4. Super-Critical Case ($q_1 > \frac{q_0}{q_0-1}$):
   The dimension $s$ is the maximal root of the simpler equation:
   $$q_0^{-s} + q_1^{-s} = 1$$
   (This corresponds to the case where the hole is empty, and the set is the entire interval).

B. Continuity Results

• Theorem 1.5: The function $(q_0, q_1) \mapsto \dim_H U_{q_0, q_1}$ is continuous for all $q_0, q_1 > 1$.
  • This extends previous results by Allaart and Kong (2019) from the single-base case ($q_0=q_1$) to the general double-base case.
• Theorem 1.4: The dimension function $d_{q_0, q_1}(a, b) = \dim_H \pi_{q_0, q_1}(\Omega_{a, b})$ is continuous with respect to the symbolic boundaries $(a, b)$.

C. Symbolic Dynamics and Entropy

• The authors establish a direct link between the topological entropy $h(\Omega_{a, b})$ and the Hausdorff dimension.
• Theorem 1.6: They prove that for certain periodic sequences $u, v$, the Hausdorff dimension is invariant under specific transformations of the boundaries (e.g., $d_{q_0, q_1}(u^\infty, v^\infty) = d_{q_0, q_1}(uv^\infty, vu^\infty)$), provided the entropy is positive. This removes technical conditions required in previous literature.

4. Significance and Impact

1. Unification of Theory: The paper unifies the study of unique expansions for single and double bases, providing a comprehensive framework that handles the asymmetry of distinct bases.
2. Explicit Computability: Unlike many fractal dimension problems where only bounds or numerical approximations are available, this work provides explicit algebraic equations (involving power series of the bases) to calculate the exact dimension.
3. Dynamical Systems Applications: The results apply broadly to open dynamical systems, including:
   • The doubling map with a hole.
   • (Linear) Lorenz maps.
   • The study of survivor sets in non-uniformly expanding systems.
4. Continuity in Parameter Space: Proving the continuity of the dimension function is a non-trivial result in fractal geometry, as dimension functions often exhibit phase transitions or discontinuities. This paper demonstrates that for double base expansions, the dimension varies smoothly with the parameters.
5. Methodological Advancement: The use of kneading invariants and the decomposition of words into "core" and "boundary" parts (via Lemma 2.3) offers a robust technique for analyzing subshifts with holes, which can be adapted to other symbolic dynamical systems.

In summary, this paper resolves the calculation of the Hausdorff dimension for double base univoque sets, establishes their continuity, and deepens the understanding of the symbolic dynamics governing these fractal sets.