For a Jordan domain with smooth boundary, the boundary rotation h is defined as the total variation of the direction angle of the tangent to the boundary curve under a complete circuit. The domain is said to have bounded turning or rotation if h < 1 and the functions under such mappings are called functions of bounded boundary turning or rotation. We define a generalization of this concept and ...

Let / be a function analytic in the unit disc, properly normalized, with bounded boundary rotation. There exists a Stieltjes integral representation for 1 +zf"(z)jf'(z). From this representation, and in view of a known variational formula for functions of positive real part, a variational formula is derived for functions of the form q(z) = l +zf"(z)jf'(z). This formula is for functions of arbit...

A conservative irrational pseudo-rotation of the two-torus is semi-conjugate to the irrational rotation if and only if it has the property of bounded mean motion [1]. (Here ‘irrational pseudo-rotation’ means a toral homeomorphism with a unique and totally irrational rotation vector.) The aim of this note is to explore this concept some further. For instance, we provide an example which shows th...

We prove that every bounded type Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This verifies Douady-Sullivan’s conjecture in the case of bounded type rotation numbers.