%0 Generic %1 levin2020renormalizations %A Levin, Genadi %D 2020 %K complex_dynamics polynomial_dynamics renormalization %T Rays to renormalizations %U http://arxiv.org/abs/2010.15386 %X Let P be a non-linear polynomial, K_P the filled Julia set of P, f a renormalization of P and K_f the filled Julia set of f. We show, loosely speaking, that there is a finite-to-one function from the set of P-external rays having limit points in K_f onto the set of f-external rays to K_f such that R and łambda(R) share the same limit set. In particular, if a point of the Julia set J_f=K_f of a renormalization is accessible from CK_f then it is accessible through an external ray of P (the inverse is obvious). Another interesting corollary is that: a component of K_P\setminus K_f can meet K_f only at a single (pre-)periodic point. We study also a correspondence induced by on arguments of rays. These results are generalizations to all polynomials (covering notably the case of connected Julia set K_P) of some results of Levin-Przytycki, Blokh-Childers-Levin-Oversteegen-Schleicher and Petersen-Zakeri where the case is considered when K_P is disconnected and K_f is a periodic component of K_P.

@preprint{levin2020renormalizations, abstract = {Let P be a non-linear polynomial, K_P the filled Julia set of P, f a renormalization of P and K_f the filled Julia set of f. We show, loosely speaking, that there is a finite-to-one function \lambda from the set of P-external rays having limit points in K_f onto the set of f-external rays to K_f such that R and \lambda(R) share the same limit set. In particular, if a point of the Julia set J_f=\partial K_f of a renormalization is accessible from C\setminus K_f then it is accessible through an external ray of P (the inverse is obvious). Another interesting corollary is that: a component of K_P\setminus K_f can meet K_f only at a single (pre-)periodic point. We study also a correspondence induced by \lambda on arguments of rays. These results are generalizations to all polynomials (covering notably the case of connected Julia set K_P) of some results of Levin-Przytycki, Blokh-Childers-Levin-Oversteegen-Schleicher and Petersen-Zakeri where the case is considered when K_P is disconnected and K_f is a periodic component of K_P.}, added-at = {2021-03-15T23:33:33.000+0100}, author = {Levin, Genadi}, biburl = {https://www.bibsonomy.org/bibtex/28928286b6049d514c626f04b6e08d682/lukas.geyer}, description = {Rays to renormalizations}, interhash = {2a93c578654f3dc04bd249be846824c0}, intrahash = {8928286b6049d514c626f04b6e08d682}, keywords = {complex_dynamics polynomial_dynamics renormalization}, note = {cite arxiv:2010.15386Comment: Minor corrections. To appear in Bull. Polish Acad. Sci. Math}, timestamp = {2021-03-15T23:33:33.000+0100}, title = {Rays to renormalizations}, url = {http://arxiv.org/abs/2010.15386}, year = 2020 }