We consider the skew product F : (x, u) → (f (x), u + τ (x)), where the base map f : T 1 → T 1 is piecewise C 2 , covering and uniformly expanding, and the fibre map τ : T 1 → R is piecewise C 2 . We show the dichotomy that either this system mixes exponentially or τ is cohomologous (via a Lipschitz function) to a piecewise constant.

For a large class of piecewise expanding C 1,1 maps of the interval we prove the Lasota-Yorke inequality with a constant smaller than the previously known 2/ inf |τ |. Consequently, the stability results of Keller-Liverani apply to this class and in particular to maps with periodic turning points. One of the applications is the stability of acim's for a class of W-shaped maps. Another application is an affirmative answer to a conjecture of Eslami-Misiurewicz regarding acim-stability of a family of unimodal maps.

Consider the skew product $F:\mathbb{T}^{2}\rightarrow \mathbb{T}^{2}$, $F(x,y)=(f(x),y+\unicode[STIX]{x1D70F}(x))$, where $f:\mathbb{T}^{1}\rightarrow \mathbb{T}^{1}$ is a piecewise $\mathscr{C}^{1+\unicode[STIX]{x1D6FC}}$ expanding map on a countable partition and $\unicode[STIX]{x1D70F}:\mathbb{T}^{1}\rightarrow \mathbb{R}$ is piecewise $\mathscr{C}^{1}$. It is shown that if $\unicode[STIX]{x1D70F}$ is not Lipschitz-cohomologous to a piecewise constant function on the joint partition of $f$ and $\unicode[STIX]{x1D70F}$, then $F$ is mixing at a stretched-exponential rate.

Keller [Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94(4) (1982), 313–333] introduced families of W-shaped maps that can have a great variety of behaviors. As a family approaches a limit W map, he observed behavior that was either described by a probability 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 neighborhoods of the fixed critical point of the limit map. In this paper, we show that the conjecture is not true. We construct a very simple family of W-maps with ACIMs 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 [P. Góra. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583].