On Discretizations of the Generalized Boole Type Transformations and their Ergodicity

The Frobenius-Perron Operator and Its Discretization:

We consider an m-dimensional; not necessary compact; C1- manifold Mm, endowed with a Lebesgue measure μ determined on the σ-algebra of Borel subsets of Mm and ϕ: Mm→Mm being an almost everywhere smooth mapping. The related Frobenius-Perron operator

is defined by means of the integral relationship

for any and all μ-measurable subsets A⊂M^m Equivalently it can be defined as a mapping on the measure space (M^m)

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