Ergodic properties of a parameterised family of symmetric golden maps: the matching phenomenon revisited

TL;DR

This paper investigates a family of symmetric golden maps, analyzing their invariant measures, the matching phenomenon, and digit frequency behavior, revealing step-function densities and continuous variation of digit frequencies with parameters.

Contribution

It provides an explicit description of invariant measures, the matching phenomenon, and digit frequency behavior for a family of parameterized maps related to the golden mean.

Findings

Invariant measures are unique and absolutely continuous.

Density functions are step functions with finitely many jumps on dense parameter subsets.

The frequency of zero digits varies continuously and reaches a maximum of 3/4.

Abstract

We study a one-parameter family of interval maps $\{T_\alpha\}_{\alpha\in[1,\beta]}$, with $\beta$ the golden mean, defined on $[-1,1]$ by $T_\alpha(x)=\beta^{1+|t|}x-t\beta\alpha$ where $t\in\{-1,0,1\}$. For each $T_\alpha,\ \alpha>1$, we construct its unique, absolutely continuous invariant measure and show that on an open, dense subset of parameters $\alpha$, the corresponding density is a step function with finitely many jumps. We give an explicit description of the maximal intervals of parameters on which the density has at most the same number of jumps. A main tool in our analysis is the phenomenon of matching, where the orbits of the left and right limits of discontinuity points meet after a finite number of steps. Each $T_\alpha$ generates signed expansions of numbers in base $1/\beta$; via Birkhoff's ergodic theorem, the invariant measures are used to determine the asymptotic relative frequencies of digits in generic $T_\alpha$-expansions. In particular, the frequency of $0$ is shown to vary continuously as a function of $\alpha$ and to attain its maximum $3/4$ on the maximal interval $[1/2+1/\beta,1+1/\beta^2]$.