A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature

Abstract

Abstract It is well known that C 2-transformation φ of the unit interval into itself with a Markov partition (2.1) π = {Ik : k ∈ K} admits φ-invariant density g (g ≥ 0, ∥g∥ = 1) if: (2.2) ∣(φn )′∣ ≥ C 1 > 1 for some n (expanding condition); (2.3) ∣φ″(x)/(φ′(y))2∣ ≤ C 2 < ∞ (second derivative condition); and (2.4) #π < ∞ or φ (Ik ) = [0, 1], for each Ik ∈ π. If (2.4) is deleted, then the situation dramatically changes. The cause of this fact was elucidated in connection with so-called Adler’s Theorem ([1] and [2]). However after that time in the literature occur claims and opinions concerning the existence of invariant densities and their properties for Markov Maps, which satisfy (2.2), (2.3) and do not satisfy (2.4), revealing unacquaintance with that question. In this note we discuss the problems arising from the mentioned claims and opinions. Some solutions of that problems are given, in a systematic way, on the base of the already published results and by providing appropriate examples.