Generalized continued fractions1

摘要

The invariant measures for the transformations x↦{φ(x)} on the unit interval are studied where (i) φ∈C2, (ii) φ′<0,(iii) φ(0+)=+∞, φ(1)=1, (iv) the map x↦{φ(x)} is eventually expansive, and (v) φ′ and φ″ satisfy some growth conditions. It is shown that the invariant ergodic measure ρ(x)dx satisfies ρ(0)=(1−1/φ′(1))·ρ(1). For φ(x)=1/xp, the limiting behavior of ρ is investigated as p→+∞ or p→p0=0.241485…