A Family of Piecewise Expanding Maps Having Singular Measure as a Limit of Acim’s

Keller [9] introduced families of W–shaped maps that can have a great variety of behavior. As a family approaches a limit W map, he observed behavior that was either described by a probabilty density function (pdf) or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (acim) can result only from the existence of small invariant neighbourhoods of the fixed critical point of the limit map. In this note we show that the conjecture is not true. We construct a very simple family of W maps with acim’s supported on the whole interval, whose limiting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [6].