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

Abstract

We study a one-parameter family of interval maps {Tα}α∈[1;β], with golden mean β the defined on [-1; 1] by Tα(x) = β1+|t|x - tβα, where t ∊ {-1; 0; 1} is determined piecewise. For each Tα; 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 Tα 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 Tα-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∕β2].