Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture.

Lyubich, Mikhail

Displaying similar documents to “Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture.”

Hyperbolicity of renormalization of critical circle maps

Michael Yampolsky (2003)

Publications Mathématiques de l'IHÉS

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Dynamics of quadratic polynomials : complex bounds for real maps

Mikhail Lyubich, Michael Yampolsky (1997)

Annales de l'institut Fourier

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We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map $z \mapsto z^{2} + c$ , $c \in [- 2, 1 / 4]$ , is locally connected.

Diffusion times and stability exponents for nearly integrable analytic systems

Pierre Lochak, Jean-Pierre Marco (2005)

Open Mathematics

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For a positive integer n and R>0, we set $B_{R}^{n} = \{x \in ℝ^{n} | {∥x∥}_{\infty} < R\}$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $h (r) = \frac{1}{2} r_{1}^{2} + . . . \frac{1}{2} r_{n - 1}^{2} + r_{n}$ on $𝕋^{n} \times B_{R}^{n}$ , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of $𝕋^{n} \times B_{R}^{n}$ , and setting $ε_{j} : = {∥h - H_{j}∥}_{C^{0} (V)}$ the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly...