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- W1616620908 abstract "The aim of the thesis is to study the renormalization of unimodal maps with low smoothness and the dynamics of H'enon renormalization. In this work we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space of C^2 unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to get a priori bounds. In this new smoother class, the failure of hyperbolicity is tamer than in C^2. Things get much worse with just a bit less of smoothness than C^2 as then even the uniqueness is lost and other asymptotic behavior become possible. Furthermore, We show that the period doubling renormalization operator acting on the space of C^{1+Lip} unimodal maps has infinite topological entropy. The second part of the thesis is devoted to the renormalization of H'enon maps. It was known that for strongly dissipative H'enon maps, there is a short curve in the parameter space which consists of infinitely renormalizable H'enon maps of period doubling type. In this work we study numerically, the extension of this curve in the parameter space up to the conservative map. More precisely, we describe the combinatorial changes which occur along this curve. Furthermore, we describe how the one-dimensional Cantor set deforms into the Cantor set of conservative map. To show this we compute the distribution of angles of the line fields along the Cantor set and explain how this geometry becomes more complicated for maps close to the infinitely renormalizable conservative maps." @default.
- W1616620908 created "2016-06-24" @default.
- W1616620908 creator A5048966120 @default.
- W1616620908 date "2008-12-01" @default.
- W1616620908 modified "2023-09-24" @default.
- W1616620908 title "Renormalization and non-rigidity" @default.
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