COMPLEX ROTATION NUMBERS XAVIER BUFF AND NATALIYA GONCHARUK

ABSTRACT. We investigate the notion of complex rotation number which was introduced by V. I. Arnold in 1978. Let f: R/Z → R/Z be an orientation pre-serving circle diffeomorphism and let ω ∈ C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary com-ponents of the annulus { z ∈C/Z | 0< Im(z)< Im(ω) } via the map f +ω. This complex torus is isomorphic to C/(Z+τZ) for some appropriate τ ∈C/Z. According to Moldavskis [5], if the ordinary rotation number rot ( f +ω0) is Diophantine and ifω tends to ω0 non tangentially to the real axis, then τ tends to rot ( f +ω0). We show that the Diophantine and non tangential assumptions are unnecessary: if rot ( f +ω0) is irrational then τ tends to rot ( f +ω0) as ω tends to ω0. This, together with results of N.Goncharuk [3], motivates us to introduce a new fractal set, given by the limit values of τ as ω tends to the real axis. For the rational values of rot ( f +ω0), these limits do not necessarily coincide with rot ( f +ω0) and form a countable number of analytic loops in the upper half-plane. Notation: • H=H+ is the set of complex numbers with positive imaginary part. • H − is the set of complex numbers with negative imaginary part. • If p/q is a rational number, then p and q are assumed to be coprime. • If x and y are distinct points in R/Z, then (x, y) denotes the set of points z ∈ R/Z − { x, y} such that the three points x, z, y are in increasing order and [x, y]: = (x, y) ∪ { x, y}. • If f: R/Z→R/Z is a circle diffeomorphism, D f:= R/Z ∣∣∣ ∣ f ′′(x)f ′(x