On Piecewise Smooth Maps of a Circle
by
Akhtam A. Dzhalilov
Mechanics and Mathematics Faculty of Samarkand State University, Boulevard St.15, Samarkand 703004, Uzbekistan

In the present work we study piecewise smooth circle homeomorphisms i.e., maps that are smooth everywhere except several singularities where the first derivative is discontinuous. We consider the orientation- preserving homeomorphism T_fx of the unit circle,

Tf(x) = ì í î f(x) ü ý þ ,     x in S1=[0,   1)

where the {·}- denotes the fractional part of a number and f(x) is a continuous strictly monotone function on R¹ with the following properties:
(a) 0 <= f(0) < 1;
(b) f(x+1)=f(x)+1 for any x in R¹;
(c) the homeomorphism T_f has the corners x_pi=T_f^p_ix₀, i = [`0, m],    p₀=0 < p₁ < p₂ < ... < p_m; furthermore, the one-sided derivatives f'(x_pi +/- 0) exist and are finite , and

f'(xpi-0) f'(xpi+0) =ci(f) =/= 1, i = 0, m

(d) f(x) in C² (S¹ \{x_pi, i = [`0, m]}),     and     f'(x) >= const > 0 for f(x) in S<sup>1</sup>\{x<sub>pi</sub>, i = [`0, m]}.

Condition (d) means that the function f(x) belongs to the class C² on any connected component of the set S¹ \{ x_pi, i = [`0, m]}.

Let the rotation number \rho = \rho(f) is irrational. Denjoy showed that if f(x) in C¹ (R¹) and var_S¹ logf'(x) < \infty, then there exists a circle homeomorphism; T_j such that T_j o T_f=T_\rho o T_j where T_\rhox={ x+\rho}, is the linear rotation of the circle through an angle of \rho. An important problem in circle maps theory is to find relations between the smoothness of f , the properties of the rotation number \rho , and the smoothness class of the conjugation T_j. This problem is closely related to the existense problem for an absolutely continuous (a.c.) invariant measure for T_f . Indeed, a unique probability invariant measure for T_f is a.c. w.r.t., the Lebesgue measure iff j(x) is an absolutely continuous function. This consideration was first used by Arnold [A] to study smoothness of j(x). The strongest results in this field were obtained by Herman, Yoccoz, Katznelson and Ornstein, Sinai and Khanin.

In turns out that the situation becomes diametrically opposite in the presence of a single point x_c of break-type singularity. Namely, if f(x) in C^{2+ \epsilon} ( S¹ \x_c ) for some \epsilon > 0, then an invariant measure is always singular w.r.t. Lebesgue measure [D]. Now we formulate our main results.

Theorem 1. Assume that a function f(x) defining a homeomorphism T_f satisfies conditions (a)-(d), \prod_i=0^m c_i =/= 1 and the rotarion number \rho = \rho(f) is irrational. Then the invariant measure \mu is singular w.r.t. the Lebesgue measure \lambda , i.e. there exists a measurable subset A subset S¹ such that \mu(A)=1 and \lambda(A)=0.

Definition 1. Two measures \nu₁ and \nu₂ on the same \sigma- algebra are L²-equivalent if \nu₁=j₁ (\nu₂) with j₁ in L² (\nu₁) and \nu₂=j₂ (\nu₁) with j₂ in L² (\nu₂).

Definition 2. Let \rho be irrational number and its continuous fraction expansion be \rho = [k₁, k₂, ..., k_n, ...]. If k_n <= const, n >= 1, then we say that \rho is of bounded type.

Theorem 2. Suppose that a function f(x) defining a homeomorphism T_f satisfies condititons (a)-(d), \prod_i=0^mc_i=1 and the rotation number \rho = \rho(f) is irrational of bounded type. Then the invariant measure \mu and the Lebesgue measure \lambda are L²-equivalent.

[A]. Arnold V.I. "Small denominators, I. Mapping the circle onto itself", Izv.Akad.Nauk SSSR, Ser.Matem., 25, 1, p. 21-86, (1961).

[D] Dzhalilov A.A. and Khanin K.M., Ön an invariant measure for homeomorphisms of a circle with a point of break", Functional Analysis and its Appl., v.32, 3, p.153-161, (1998).

Date received: February 24, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calu-91.