On analytic perturbations of a family of Feigenbaum-like equations
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| Title: | On analytic perturbations of a family of Feigenbaum-like equations |
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| Authors: | Denis Gaidashev |
| Source: | Journal of Mathematical Analysis and Applications. 374:355-373 |
| Publication Status: | Preprint |
| Publisher Information: | Elsevier BV, 2011. |
| Publication Year: | 2011 |
| Subject Terms: | Renormalization, A-priori bounds, Mathematics - Complex Variables, Applied Mathematics, Composition operators, Dynamical Systems (math.DS), 01 natural sciences, Period-doubling, Herglotz functions, 0103 physical sciences, Unimodal maps, FOS: Mathematics, Mathematics - Dynamical Systems, Complex Variables (math.CV), 37E20, 30C20, 47B33, 0101 mathematics, Analysis |
| Description: | We prove existence of solutions $(��,��)$ of a family of of Feigenbaum-like equations \label{family} ��(x)={1+\eps \over ��} ��(��(��x)) -\eps x +��(x), where $\eps$ is a small real number and $��$ is analytic and small on some complex neighborhood of $(-1,1)$ and real-valued on $\fR$. The family $(\ref{family})$ appears in the context of period-doubling renormalization for area-preserving maps (cf. \cite{GK}). Our proof is a development of ideas of H. Epstein (cf \cite{Eps1}, \cite{Eps2}, \cite{Eps3}) adopted to deal with some significant complications that arise from the presence of terms $\eps x +��(x)$ in the equation $(\ref{family})$. The method relies on a construction of novel {\it a-priori} bounds for unimodal functions which turn out to be very tight. We also obtain good bounds on the scaling parameter $��$. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function. |
| Document Type: | Article |
| Language: | English |
| ISSN: | 0022-247X |
| DOI: | 10.1016/j.jmaa.2010.06.047 |
| DOI: | 10.48550/arxiv.0811.2821 |
| Access URL: | http://arxiv.org/abs/0811.2821 https://www.sciencedirect.com/science/article/abs/pii/S0022247X10005500 https://core.ac.uk/display/82157170 https://www.sciencedirect.com/science/article/pii/S0022247X10005500 http://www.diva-portal.org/smash/record.jsf?pid=diva2:375477 http://swepub.kb.se/bib/swepub:oai:DiVA.org:uu-135342 http://www2.math.uu.se/~gaidash/Papers/Feigenbaum_Pert.pdf |
| Rights: | Elsevier Non-Commercial arXiv Non-Exclusive Distribution |
| Accession Number: | edsair.doi.dedup.....6eb2dadd7a9b1dc52315a504da4afe2f |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | We prove existence of solutions $(��,��)$ of a family of of Feigenbaum-like equations \label{family} ��(x)={1+\eps \over ��} ��(��(��x)) -\eps x +��(x), where $\eps$ is a small real number and $��$ is analytic and small on some complex neighborhood of $(-1,1)$ and real-valued on $\fR$. The family $(\ref{family})$ appears in the context of period-doubling renormalization for area-preserving maps (cf. \cite{GK}). Our proof is a development of ideas of H. Epstein (cf \cite{Eps1}, \cite{Eps2}, \cite{Eps3}) adopted to deal with some significant complications that arise from the presence of terms $\eps x +��(x)$ in the equation $(\ref{family})$. The method relies on a construction of novel {\it a-priori} bounds for unimodal functions which turn out to be very tight. We also obtain good bounds on the scaling parameter $��$. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function. |
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| ISSN: | 0022247X |
| DOI: | 10.1016/j.jmaa.2010.06.047 |