A remark on Denjoy's inequality for PL circle homeomorphisms with two break points

Abstract

It is well known that for a P-homeomorphism f of the circle S1=R/Z with irrational rotation number ρf the Denjoy's inequality |log⁡Dfqn|≤V holds, where V is the total variation of log⁡Df and qn, n≥1, are the first return times of f. Let h be a piecewise-linear (PL) circle homeomorphism with two break points a0, c0, irrational rotation number ρh and total jump ratio σh=1. Denote by Bn(h) the partition determined by the break points of hqn and by μh the unique h-invariant probability measure. It is shown that the derivative Dhqn is constant on every element of Bn(h) and takes either two or three values. Furthermore we prove, that log⁡Dhqn can be expressed in terms of μh-measures of some intervals of the partition Bn(h) multiplied by the logarithm of the jump ratio σh(a0) of h at the break point a0.