Perron–Frobenius operators acting on BV(I) as contractors

Abstract

Let BV(I) be the (Banach) space of real functions h on I:=[0,1] of bounded variation \mathop{\rm var}\nolimits h with the norm \Vert h\Vert = \vert h\vert + \mathop{\rm var}\nolimits h, where \vert h\vert :=\sup_{t\in I}\vert h(t)\vert. We introduce a condition (M) on the monotony of the transition probability function (TPF) of the piecewise monotonic transformation \tau associated with an f-expansion. The transfer operator associated with \tau is denoted by U. Under (M) we can apply methods used in the continued fraction expansion case to U acting on BV(I). An expression in terms of the TPF of a contraction constant, \theta, of \mathop{\rm var}\nolimits Uh is thus obtained which is subsequently used to state a Gauss–Kuzmin–Levy theorem. This result yields the answer in a special case to the problem of determining a numerical value of the constant \theta occurring in the Kuzmin-type theorem proved in [1].

Examples of numerical values of \theta in particular cases are given. Possible extensions of the methods used are mentioned.