Abstract

We present some results on the existence of absolutely continuous invariant measures (acim's) using the Fourier-Stieltjes transform. We consider the sequence of Perron Frobenius operators $$\{f,P\sb{\tau}f,P\sbsp{\tau}{2}f,P\sbsp{\tau}{3}f,\... ,P\sbsp{\tau}{n}f,\...\},$$induced by the nonsingular transformation $\tau:I\to I,$ with $f \in {\cal L}\sp1$, and the associated sequence of Fourier-Stieltjes transforms$$\{ {\cal F}(F\sb0), {\cal F}(F\sb1), {\cal F}(F\sb2),\...,{\cal F}(F\sb{n}),\...\},$$where $F\sb{n}(x) = f\sbsp{-\infty}{x}\ P\sbsp{\tau}{n}f(u)du$ and ${\cal F}(F\sb{n})(t) = f\sbsp{-\infty}{\infty}\ e\sp{itx}dF\sb{n}(x).$ The main result is: if $\tau$ is piecewise monotonic expanding and ${1\over{{d\over dx}\tau(\cdot)}}$ is a function of bounded variations, then $\tau$ has an acim. Although this is a known result, the method of proof is new and may allow generalizations needed. Finally, we introduce criteria on the Fourier Stieltjes Transforms needed to ensure the existence of acim's.

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