Abstract
For an infinitely renormalizable quadratic map fc: z2 + c with the sequence of renormalization periods {km} and rotation numbers {tm = pm/qm}, we prove that if lim sup km -1 log {pipe} pm{pipe} > 0, then the Mandelbrot set is locally connected at c. We prove also that if lim sup {pipe}tm+1{pipe}1/qm < 1 and qm → ∞, then the Julia set of fc is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor.
Original language | English |
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Pages (from-to) | 295-328 |
Number of pages | 34 |
Journal | Communications in Mathematical Physics |
Volume | 304 |
Issue number | 2 |
DOIs | |
State | Published - Jun 2011 |
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics