# Invariant measures, matching and the frequency of 0 for signed binary expansions

## TL;DR

This paper studies a family of symmetric doubling maps generating signed binary expansions, analyzing how the frequency of zero digits varies with the parameter, and provides explicit formulas and structural insights into the invariant measures and digit frequencies.

## Contribution

It introduces a parametrized family of maps with explicit invariant measures, characterizes their smoothness, and derives formulas for digit frequency dependence on parameters.

## Key findings

- Frequency of zero digits depends continuously on the parameter.
- The density of invariant measures is piecewise smooth except on a measure-zero set.
- The digit zero frequency equals 2/3 only on a specific parameter interval.

## Abstract

We introduce a parametrised family of maps $\{S_{\eta}\}_{\eta \in [1,2]}$, called symmetric doubling maps, defined on $[-1,1]$ by $S_\eta (x)=2x-d\eta$, where $d\in \{-1,0,1 \}$. Each map $S_\eta$ generates binary expansions with digits $-1$, 0 and 1. We study the frequency of the digit 0 in typical expansions as a function of the parameter $\eta$. The transformations $S_\eta$ have a natural ergodic invariant measure $\mu_\eta$ that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure $\mu_{\eta}([-\frac12,\frac12])$ by the Ergodic Theorem. We show that the density of $\mu_\eta$ is piecewise smooth except for a set of parameters of zero Lebesgue measure and full Hausdorff dimension and give a full description of the structure of the maximal parameter intervals on which the density is piecewise smooth. We give an explicit formula for the frequency of the digit 0 in typical signed binary expansions on each of these parameter intervals and show that this frequency depends continuously on the parameter $\eta$. Moreover, it takes the value $\frac23$ only on the interval $\big[ \frac65, \frac32\big]$ and it is strictly less than $\frac23$ on the remainder of the parameter space.