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

Góra, P.; Boyarsky, A.

Can. J. Math. 41, No. 5, 855-869 (1989).

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

Keywords:

fixed points; Frobenius-Perron operator; absolutely continuous invariant measures; invariant density; compactness; sequence of piecewise linear maps; non-expanding maps

Citations:

Zbl 0255.28018