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

Abstract

We study a one-parameter family of interval maps $\{T_α\}_{α\in[1,β]}$, with $β$ the golden mean, defined on $[-1,1]$ by $T_α(x)=β^{1+|t|}x-tβα$ where $t\in\{-1,0,1\}$. For each $T_α,\ α>1$, we construct its unique, absolutely continuous invariant measure and show that on an open, dense subset of parameters $α$, 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 $T_α$ generates signed expansions of numbers in base $1/β$; via Birkhoff's ergodic theorem, the invariant measures are used to determine the asymptotic relative frequencies of digits in generic $T_α$-expansions. In particular, the frequency of $0$ is shown to vary continuously as a function of $α$ and to attain its maximum $3/4$ on the maximal interval $[1/2+1/β,1+1/β^2]$.