An inverse problem in Pólya--Schur theory. II. Exactly solvable operators and complex dynamics
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909412272111616 |
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| author | Alexandersson, Per Hemmingsson, Nils Shapiro, Boris |
| author_facet | Alexandersson, Per Hemmingsson, Nils Shapiro, Boris |
| contents | This paper, being the sequel of [An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators], studies a class of linear ordinary differential operators with polynomial coefficients called \emph{exactly solvable}; such an operator sends every polynomial of sufficiently large degree to a polynomial of the same degree. We focus on invariant subsets of the complex plane for such operators when their action is restricted to polynomials of a fixed degree and discover a connection between this topic and classical complex dynamics and its multi-valued counterpart. As a very special case of invariant sets we recover the Julia sets of rational functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01643 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An inverse problem in Pólya--Schur theory. II. Exactly solvable operators and complex dynamics Alexandersson, Per Hemmingsson, Nils Shapiro, Boris Dynamical Systems 37F10 (Primary), 34A30 (Secondary) This paper, being the sequel of [An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators], studies a class of linear ordinary differential operators with polynomial coefficients called \emph{exactly solvable}; such an operator sends every polynomial of sufficiently large degree to a polynomial of the same degree. We focus on invariant subsets of the complex plane for such operators when their action is restricted to polynomials of a fixed degree and discover a connection between this topic and classical complex dynamics and its multi-valued counterpart. As a very special case of invariant sets we recover the Julia sets of rational functions. |
| title | An inverse problem in Pólya--Schur theory. II. Exactly solvable operators and complex dynamics |
| topic | Dynamical Systems 37F10 (Primary), 34A30 (Secondary) |
| url | https://arxiv.org/abs/2412.01643 |