Abstract

The family of $W-$shaped maps was introduced in 1982 by G. Keller. Based on the investigation of the properties of the maps, it was 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 limiting map. We show that the conjecture is not true by constructing a family of $W-$shaped maps with acims supported on the entire interval, whose limiting dynamical behavior is described by a singular
measure. We then generalize the above result by constructing families of $W-$shaped maps $\{W_a\}$ with a turning fixed point having slope $s_1$ on one side and $-s_2$ on the other. Each such $W_a$ map has an acim $\mu_a$. Depending on whether $\frac{1}{s_1}+\frac{1}{s_2}$ is larger, equal, or smaller than 1, we show that the limit of $\mu_a$ is a singular measure, a combination of singular and absolutely continuous measure or an acim, respectively. We also consider $W$-shaped maps satisfying $\frac 1{s_1}+\frac 1{s_2}=1$ and show that the eigenvalue $1$ of the associated Perron-Frobenius operator is not stable, which in turn implies the instability of the isolated spectrum. Motivated by the above results, we introduce the harmonic average of slopes condition, with which we obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Moreover, we prove stability results using Rychlik's Theorem together with the harmonic average of slopes condition for piecewise expanding maps.