We describe the way in which the sign of the Schwarzian derivative for a family of diffeomorphisms of the interval I affects the dynamics of an associated many-to-one skew product map of the cylinder (R/Z) × I.

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of the Schwarzian derivative, exploiting...

The essential operation that characterizes most power electronics circuits is the cyclic switching of the circuit con guration from one linear system to another. Such periodic switching operation naturally permits discrete-time modeling of power electronics systems in the form of iterative functions. In this section we review iterative functions of rst order and in particular discuss a techniqu...

A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in Cn are introduced. Basic properties such as the chain rule, multiplicative invariance affine proved these operators. It is shown that stable only with respect to rotations of identity. characterization given case when holomorphic. Furthermore, it if mapping vanishes then analytic part this Möbius transf...

Abstrac t . We consider several results, each o f which uses some type o f " L 2' ' es t imate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points o f a curve in terms of a certain geometric square function. Our next result is an LP est imate relating the derivative of a conformal mapping to its Schwarzian derivative. One c...