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

Abstract

It is well known that C2-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)′∣ ≥ C1 > 1 for some n (expanding condition); (2.3) ∣φ″(x)/(φ′(y))2∣ ≤ C2 < ∞ (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]).

Keywords

• markov maps
• invariant measure
• expanding transformation
• recurrence
• aperiodicity
• 28d05
• 37a99
• 26a18