$W$-like maps with various instabilities of acim's

TL;DR

This paper constructs a family of W-like maps with a turning fixed point to study how their absolutely continuous invariant measures behave under different slope conditions, revealing various types of measure limits.

Contribution

It generalizes previous results by analyzing the limit behavior of invariant measures for W-like maps with different slope conditions and provides a counterexample regarding invariant density bounds.

Findings

Limit measure can be singular, mixed, or absolutely continuous depending on slopes.

Invariant densities may lack a positive lower bound even for expanding maps.

The behavior depends on the sum of reciprocals of slopes.

Abstract

This paper generalizes the results of [13] and then provides an interesting example. We construct a family of $W$-like maps $\{W_a\}$ with a turning fixed point having slope $s_1$ on one side and $-s_2$ on the other. Each $W_a$ has an absolutely continuous invariant measure $\mu_a$. 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 absolutely continuous measure, respectively. It is known that the invariant density of a single piecewise expanding map has a positive lower bound on its support. In Section 4 we give an example showing that in general, for a family of piecewise expanding maps with slopes larger than 2 in modulus and converging to a piecewise expanding map, their invariant densities do not necessarily have a positive lower bound on the support.