. D S ] 2 A ug 2 00 6 GENERIC EXPANDING MAPS WITHOUT ABSOLUTELY CONTINUOUS INVARIANT σ - FINITE MEASURE
نویسنده
چکیده
If f is a measurable transformation of a Lebesgue measure space (X,A, λ) to itself, that does not preserve the measure λ, one can study the invariant measures of f and compare them to λ. A especially interesting case is when f is non-singular with respect to λ (in the sense thatλ(A) = 0 iff λ( f(A)) = 0), but nevertheless there exist no σ-finite invariant measure which is absolutely continuous with respect to λ. Such maps f are called of type III (with respect to the measure). Their existence was conjectured by Halmos [H] and established by Ornstein [O]. Other examples were given later; let us cite a few (when not specified, the relevant measure is Riemannian):
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