LAUSR

Abstract It is well known that for a P-homeomorphism f of the circle S 1 = R / Z with irrational rotation number ρ f the Denjoy's inequality | log ⁡ D f q n | ≤ V holds, where V is the total variation of log ⁡ D f and q n , n ≥ 1 , are the first return times of f. Let h be a piecewise-linear (PL) circle homeomorphism with two break points a 0 , c 0 , irrational rotation number ρ h and total jump ratio σ h = 1 . Denote by B n ( h ) the partition determined by the break points of h q n and by μ h the unique h-invariant probability measure. It is shown that the derivative D h q n is constant on every element of B n ( h ) and takes either two or three values. Furthermore we prove, that log ⁡ D h q n can be expressed in terms of μ h -measures of some intervals of the partition B n ( h ) multiplied by the logarithm of the jump ratio σ h ( a 0 ) of h at the break point a 0 .