Invariant Measures for Certain Linear Fractional Transformations Mod 1

Explicit invariant measures are derived for a family of finite-toone, ergodic transformations of the unit interval having indifferent periodic orbits. Examples of interesting, non-trivial maps of [0, 1] for which one can readily compute an invariant measure absolutely continuous to Lebesgue measure are not easy to come by. The familiar examples are the Gauss map, the backward continued fraction map, and other very special cases which are close in form to the first two. See [1], [3], and [4] for an overview of the literature. Maps for which the invariant measure is infinite are even less in evidence. We will consider a family of mappings Tk,n of the unit interval that are essentially finite-to-one analogues of the backward continued fraction maps Tk = 〈 1 uk(1−x) 〉 studied in [4], where uk = 4 cos π k+2 and 〈x〉 is the fractional part of x. Both Tk and Tk,n are Möbius transformations mod 1 having indifferent periodic orbits of period k containing zero. Surprisingly, for fixed k, Tk,n converges uniformly to Tk on compact subsets of [0, 1). An explicit formula will be given for a Tk,n-invariant measure that is absolutely continuous to Lebesgue measure on [0, 1]. The measure is infinite and the density ρk,n is C∞ in the complement of the indifferent periodic orbit. Also, ρk,n converges to the Tk-invariant density ρk derived in [4]. Using Thaler’s analysis in [6] of mappings of [0, 1] with indifferent fixed points, it is possible to show that the maps Tk,n are ergodic with respect to Lebesgue measure. From now on suppose that k > 0 and n > 1 in N have been fixed and that if k = 1, then n > 2. We begin by defining the Möbius transformation that will determine Tk,n. Write Aα(x) = −αx− nα + α (n− α− 1)x− n + 1 . Then Aα(0) = α, Aα(1) = n, and A−1 α (x) = (n−1)x−nα+α (n−α−1)x+α . Aα has the following properties. Received by the editors January 1, 1998. 1991 Mathematics Subject Classification. Primary 11J70, 58F11, 58F03.