On the geometry of hedgehogs and log-Riemann surfaces

We study three related classes of objects arising in Complex Dynamics and Riemann Surfaces, namely hedgehogs, log-Riemann surfaces and tube-log Riemann surfaces.; Hedgehogs, which were first discovered by R. Perez-Marco, are invariant continua for holomorphic diffeomorphisms near irrationally indifferent fixed points. Heuristically one thinks of them as degenerate linearisation domains. Perez Marco's work has shown that the topological structure of hedgehogs is complex. We construct explicit examples of dynamics and hedgehogs with specific geometries. We construct hedgehogs of Hausdorff dimension one, and hedgehogs containing smooth "combs" (sets homeomorphic to the product of a Cantor set and an interval).; For these constructions we use the techniques of tube-log Riemann surfaces due to Perez-Marco. These are Riemann surfaces formed by isometric cutting and pasting of complex planes and complex cylinders along slits. Log-Riemann surfaces are a subclass of such surfaces, formed by pasting only complex planes together. In joint work with Perez-Marco, we develop and study the theory of tube-log and log-Riemann surfaces.; The theory of log-Riemann surfaces is studied in three parts, namely their geometric, analytic and algebraic theories. We first formalize the notion of log-Riemann surface, and other basic notions such as those of finite and infinite ramification points.; We discuss briefly the type problem for log-Riemann surfaces. We introduce a notion of convergence for log-Riemann surfaces and domains in log-Riemann surfaces, generalizing Caratheodory's Kernel Convergence. Log-Riemann surfaces come equipped with a distinguished set of charts, which allow one to express their uniformizations by formulae. This establishes a link between transcendental functions and geometries, which we study in detail for the class of log-Riemann surfaces with a finite number of infinite ramification points.; Uniformization formulae for these surfaces allow us to define a ring of special functions on a surface, and reconstruct algebraically all points, including infinite ramification points, as maximal ideals of this ring.; For a generic class of rational functions, we show how to construct tube-log Riemann surfaces whose uniformizations are given by the primitives of these rational functions.