TY - JOUR
T1 - Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
AU - Levin, Genadi
N1 - Funding Information:
Research supported in part by an ISF grant number 799/08.
PY - 2011/6
Y1 - 2011/6
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=79955771918&partnerID=8YFLogxK
U2 - 10.1007/s00220-011-1228-7
DO - 10.1007/s00220-011-1228-7
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AN - SCOPUS:79955771918
SN - 0010-3616
VL - 304
SP - 295
EP - 328
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 2
ER -