## Compactness of invariant densities for families of expanding, piecewise monotonic transformations.(English)Zbl 0689.28007

If $$I=[0,1]$$, m $$=$$ Lebesgue measure, then it is well-known (Lasota and Yorke) that fixed points of the Frobenius-Perron operator $P_{\tau}:\quad L_ 1(I,m)\to L_ 1(I,m),\quad P_{\tau}f(x)=\frac{d}{dx}\int_{\tau^{-1}[0,x]}f(s)dm(s),$ correspond to absolutely continuous invariant measures for the non- singular transformatin $$\tau:\quad I\to I.$$
The question of computing an invariant density $$f^*$$ (i.e. satisfying $$P_{\tau}f^*=f^*$$) seems to be exceedingly difficult, even practical procedures for finding approximations to $$f^*$$ seem to be unknown.
The main objective of this paper is to give a correct version of the results of A. A. Kosjakin and E. A. Sandler [Izv. Vyssh. Uchebn. Zaved., Mat. 3(118), 32-40 (1972; Zbl 0255.28018)] where a general procedure is outlined for finding approximation to invariant densities. However there are major gaps in their arguments and there are errors. In section 2 a compactness result is given which is used to proved the existence of sequence of piecewise linear maps whose densities converge to the invariant density of $$\tau$$. The ideas of this paper are carried over to non-expanding maps in Jour. Stat. Phys. 51, 179-194 (1988) by the authors and H. Proppe.
Reviewer: G.R.Goodson

### MSC:

 28D05 Measure-preserving transformations 37A99 Ergodic theory

Zbl 0255.28018
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