Importance of bounded variation condition for stability of an absolutely continuous invariant measure (Q1323174)

It is known that under certain conditions, a piecewise expanding map \(\tau\) on an interval \([a,b]\) admits an invariant measure absolutely continuous with respect to Lebesgue measure. The main tool is the Perron- Frobenius operator \(P_ \tau\) on \(L^ 1([a,b],m)\), where \(m\) is Lebesgue measure. A function which satisfies \(P_ \tau f= f\) is the density of an absolutely continuous invariant measure. It is extremely difficult to find such a density function for such \(\tau\). The author and \textit{A. Boyarsky} [Can. J. Math. 41, No. 5, 855-869 (1989; Zbl 0689.28007); Proc. Am. Math. Soc. 105, No. 4, 922-928 (1989; Zbl 0678.28008)] proved a compactness theorem which gives rise to a method of approximating such functions \(f\) by a sequence \(f_ n\to f\) in the \(L^ 1\)-norm, and which may more readily be calculated. In this paper, an example is constructed showing that the uniformly bounded variation condition in the compactness theorem is crucial for such approximations, and omitting it may give rise to incorrect results.