Invariant measure for double base expansions

Abstract: Given a pair $Q=(q_0,q_1)\in(1,\infty)<sup>2$ with $q_0+q_1\ge q_0q_1$, a sequence $(c_i)\in\set{0,1}<sup>\infty$ is called a $Q$-expansion of $x$ if<br/>\begin{equation*}<br/>x=\sum_{i=1}^{\infty}\frac{c_i}{q_{c_1}\cdots q_{c_i}}.<br/>\end{equation*}<br/>We primarily study the dynamical properties of the greedy and lazy maps, which are the piecewise-linear maps on the interval $I_Q=[0,\,1/(q_1-1)]$ defined by the corresponding algorithms for $Q$-expansions. <br/>We show that the greedy and lazy maps each of which has a unique absolutely continuous invariant probability measure, equivalent to the Lebesgue measure on the intervals<br/>\begin{equation*}<br/>\left[0,\frac{q_0}{q_1}\right)\qtq{and}\left(\frac{q_1}{q_0(q_1-1)}-1,\frac{1}{q_1-1}\right],<br/>\end{equation*}<br/>respectively. <br/>Furthermore, the corresponding dynamical systems are exact on $I_Q$. <br/>As a dynamical consequence, under the stronger condition $q_0+q_1&gt;q_0q_1$ the set of points having unique $Q$-expansions has Lebesgue measure zero, and almost every $x\in I_{Q}$ admits a continuum of $Q$-expansions.

• The paper establishes the existence of unique absolutely continuous invariant probability measures for greedy and lazy double-base expansions using Perron-Frobenius theory and the Lasota-Yorke framework.
• It shows that both the greedy and lazy maps are exact, mixing, and ergodic, confirming strong stochastic behavior in these dynamical systems.
• The research proves that almost every number has multiple double-base expansions while unique expansions form a set of Lebesgue measure zero.

Invariant Measures and Dynamical Properties in Double Base Expansions

Overview

This paper addresses the ergodic theoretic and measure-theoretic properties of expansions of real numbers in double-base systems, or $(q_0,q_1)$-expansions, where $q_0, q_1 > 1$ and $q_0+q_1\geq q_0 q_1$. The focus is on the dynamical systems generated by the greedy and lazy algorithms—canonical representatives of maximal and minimal lexicographical expansions—in this context. The main results demonstrate the existence, explicit structure, and statistical properties of unique absolutely continuous invariant probability measures (ACIPMs) for the greedy and lazy maps, their equivalence to Lebesgue measure on specified intervals, and the exactness of the corresponding dynamical systems. Theoretical implications for the uniqueness of expansions are also derived, connecting to the classical theory of $q$-expansions in single and multiple bases.

Double-Base Expansions and Dynamical Systems

Let $Q=(q_0, q_1)$, and for $x \in I_Q = [0, 1/(q_1-1)]$, a $Q$-expansion is a sequence $(c_i)\in\{0,1\}^\mathbb{N}$ such that

$$x = \sum_{i=1}^{\infty} \frac{c_i}{q_{c_1}\cdots q_{c_i}}.$$

Assuming $q_0+q_1\geq q_0 q_1$, every $q_0, q_1 > 1$0 has at least one such expansion, and the set $q_0, q_1 > 1$1 is invariant under the relevant transformations. The greedy map $q_0, q_1 > 1$2 and lazy map $q_0, q_1 > 1$3 are defined as piecewise-affine transformations derived from the digit selection mechanisms of the corresponding expansion algorithms.

Absolutely Continuous Invariant Measures

A central result establishes that both the greedy and lazy maps admit unique ACIPMs, denoted $q_0, q_1 > 1$4 and $q_0, q_1 > 1$5, which are mutually singular in general and are each equivalent to Lebesgue measure on their supports: $q_0, q_1 > 1$6 for the greedy map ($q_0, q_1 > 1$7), and $q_0, q_1 > 1$8 for the lazy map ($q_0, q_1 > 1$9). This is rigorously obtained by conjugating the interval $q_0+q_1\geq q_0 q_1$0 to $q_0+q_1\geq q_0 q_1$1, employing the Lasota-Yorke framework for piecewise expanding maps [18,19], and invoking Perron-Frobenius theory to yield explicit jump function formulas for invariant densities. The densities are shown to be bounded, with $q_0+q_1\geq q_0 q_1$2 non-increasing and $q_0+q_1\geq q_0 q_1$3 non-decreasing.

Exactness and Strong Stochasticity

Both the greedy and lazy dynamical systems are shown to be exact, and thus also mixing and ergodic in the measure-theoretic sense. Exactness implies rapid loss of memory of initial conditions and maximal stochasticity with respect to the invariant measure, solidifying the connection between digit expansion in double base systems and strong random-like dynamical behavior.

Properties of Unique Expansions

An important theoretical implication is the measure-theoretic size of the set of numbers with unique $q_0+q_1\geq q_0 q_1$4-expansions. Under the strict inequality $q_0+q_1\geq q_0 q_1$5, this univoque set $q_0+q_1\geq q_0 q_1$6 has Lebesgue measure zero, paralleling the single-base case. Moreover, almost every $q_0+q_1\geq q_0 q_1$7 admits a continuum of distinct $q_0+q_1\geq q_0 q_1$8-expansions, a fact deduced via ergodic and Birkhoff-type limit theorems for the $q_0+q_1\geq q_0 q_1$9-expansion process.

Explicit Invariant Density Representation

Proportional explicit representations for the invariant density are given in terms of jump functions involving the greedy and lazy expansions of critical points (the endpoints of $q$0). The density for the greedy algorithm, $q$1, is expressed as a sum over partitions determined by the expansion structure, and similarly for $q$2. In certain degenerate cases (e.g., $q$3), these densities are constant and coincide with the normalized Lebesgue density.

Theoretical and Practical Implications

From a theoretical perspective, these results provide a comprehensive framework for understanding measure-theoretic and dynamical properties in multi-base and non-uniform numeration systems, extending classical results by Rényi, Parry, and Lasota-Yorke to double-base contexts. The exactness and ACIPM structure are essential for statistical properties of digit sequences, dimension theory, and further arithmetic applications, such as normality and entropy calculations.

Practically, these findings inform algorithmic construction of random and pseudo-random number generators, numerical systems with redundancy, and coding schemes, especially where non-integer or multiple bases arise (such as in fractal geometry or non-uniform quantization).

Conclusion

This paper rigorously characterizes the invariant measure structure and strong stochastic properties of digit expansion algorithms in double base numeration systems. It establishes the existence and explicit structure of unique ACIPMs for both greedy and lazy maps, the exactness of the induced dynamical systems, and proves that unique expansions are a null phenomenon in this context. These results solidify the connection between symbolic dynamics, ergodic theory, and number theory for double-base representations, suggesting further exploration into higher-dimensional or more general non-uniform numeration schemes and their associated dynamical systems.

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References:

• Lasota, A., Yorke, J. A., 1982. "Exact dynamical systems and the Frobenius-Perron operator." Trans. Amer. Math. Soc. 273, no. 1: 375-384.
• Parry, W., 1960. "On the β-expansions of real numbers." Acta Math. Acad. Sci. Hungar. 11: 401-416.
• Rényi, A., 1957. "Representations for real numbers and their ergodic properties." Acta Math. Acad. Sci. Hungar. 8: 477-493.

See "Invariant measure for double base expansions" (2605.08641).