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Among Lyubich's fundamental results in one-dimensional dynamics is his proof in the 1990s of hyperbolicity of renormalization for unimodal maps (conjectured by Feigenbaum and by Coullet and Tresser in the 1970s). Renormalization has been one of the main themes in low-dimensional dynamics for the past 40 years. Sullivan and later McMullen proved parts of the renormalization picture for unimodal maps, and Lyubich completed the proof of universality for bounded combinatorics. He later constructed a ``full hyperbolic horseshoe'' for the renormalization operator acting on real quadratic-like maps.
In earlier work on rigidity of quadratic polynomials, Lyubich resolved perhaps the most famous problem in dynamics on the real line by showing that hyperbolicity is dense in the real quadratic family. (This result was independently obtained by J. Graczyk and G. Świątek.)
One of the most fundamental problems in dynamics, for a parameterized family of maps, is to understand the asymptotic behaviour of almost every orbit for almost every value of the parameter. Even for the family of quadratic interval maps, this question had eluded experts for years. Lyubich's construction of the full renormalization horseshoe, together with his joint work with M. Martens and T. Nowicki, allowed him to obtain the definitive answer: almost every quadratic map is either regular or stochastic.
Lyubich's work was a major step towards the celebrated MLC (Mandelbrot set is locally connected) conjecture. A series of new breakthroughs has come in his recent results with J. Kahn, using the Kahn-Lyubich quasi-additivity law in conformal geometry.
In earlier work on rigidity of quadratic polynomials, Lyubich resolved perhaps the most famous problem in dynamics on the real line by showing that hyperbolicity is dense in the real quadratic family. (This result was independently obtained by J. Graczyk and G. Świątek.)
One of the most fundamental problems in dynamics, for a parameterized family of maps, is to understand the asymptotic behaviour of almost every orbit for almost every value of the parameter. Even for the family of quadratic interval maps, this question had eluded experts for years. Lyubich's construction of the full renormalization horseshoe, together with his joint work with M. Martens and T. Nowicki, allowed him to obtain the definitive answer: almost every quadratic map is either regular or stochastic.
Lyubich's work was a major step towards the celebrated MLC (Mandelbrot set is locally connected) conjecture. A series of new breakthroughs has come in his recent results with J. Kahn, using the Kahn-Lyubich quasi-additivity law in conformal geometry.
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arxiv(2024)
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arXiv (Cornell University) (2023)
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GEOMETRIC AND FUNCTIONAL ANALYSISno. 4 (2023): 912-1047
Arnold Mathematical Journalno. 4 (2023): 505-597
arXiv (Cornell University) (2023)
CONFORMAL GEOMETRY AND DYNAMICSno. 1 (2023): 1-54
arXiv (Cornell University) (2023)
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