## Beyond hyperbolicity: expansion properties of one dimensional mappingsMathematical Sciences Institute, Cornell University, 1990 - 22 pages |

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90-86 BEYOND HYPERBOLICITY absolutely continuous invariant absorbing Cantor sets attracting Cantor set bounded independent chain rule formula class of S-unimodal Conjugacies between unimodal constructed continuous invariant measure Cornell University critical point critical value dark realm dependence to initial derivative along trajectories diffeomorphisms DIMENSIONAL MAPPINGS dynamical systems equivalent S-unimodal maps existence of absolutely EXPANSION PROPERTIES exponential finite union fixed point funnel geometric growth rate Hausdorff dimension homeomorphism indifferent period orbit infinitely renormalizable initial conditions interval Jp itineraries John Guckenheimer metric attractor metric structure number of iterates orbit of period parameter families parameter values giving periodic point point at xo positive Lyapunov exponent quasisymmetric reduce any growth renormalization satisfy Axiom Schwarzian derivative sensitive dependence set of parameter stable periodic orbits Stewart Johnson stopping rules structurally stable structurally unstable Theorem topological attractors topological equivalence classes topologically equivalent S-unimodal transformations of Misiurewicz transformations with absolutely type III example unimodal transformations values giving transformations weaker obstruction