Mathematical Sciences: The Method of Induced Hyperbolicity for One-Dimensional Maps and Two-Dimensional Diffeomorphisms

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Using the method of induced hyperbolicity developed earlier by the principal investigator, families of nonlinear dynamical systems will be studied. Of particular interest will be their stochastic behavior. One-parameter families of diffeomorphisms close to the unstable manifold of the Feigenbaum fixed point will be inspected. And Sullivan's approach will be used to investigate when topological conjugacy implies quasiconformal conjugacy for maps with absolutely continuous invariant measures. An invariant measure of a dynamical system is positive on invariant sets of the system which attract or repel. The principal investigator will study such systems which are close in some parameter space to the Feigenbaum limit. This limit of period-doubling bifurcations has raised intense interest from investigators in a wide variety of physical and biological sciences. Just past this limit lies the onset of chaos.