Approach to equilibrium for locally expanding maps in ℝk

By using a well known technique from classical statistical mechanics of one-dimensional lattice spin systems we prove existence of an absolutely continuous invariant asymptotic measure for certain locally expanding mapsT of the unit cube in ℝk. We generalize herewith in a certain sense the results of Lasota and Yorke on piecewise expanding maps of the unit interval to higher dimensions. We show a Kuzmin-type theorem for these systems from which exponential approach to equilibrium and strong mixing properties follow.

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