Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914049852178432 |
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| author | Ji, Zhuchao Xie, Junyi |
| author_facet | Ji, Zhuchao Xie, Junyi |
| contents | The aims of this paper are to answer several conjectures and questions about multiplier spectrum of rational maps and to give new proofs of several rigidity theorems in complex dynamics, by combining tools from complex and non-archimedean dynamics.
A remarkable theorem due to McMullen asserts that aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston's rigidity theorem for post-critically finite maps, in which Teichmüller theory is an essential tool. We will give a new proof of McMullen's theorem without using quasiconformal maps or Teichmüller theory.
We show that aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen's theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik.
We show that a rational map is exceptional if and only if one of the following holds (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field; (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_13258 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics Ji, Zhuchao Xie, Junyi Dynamical Systems Algebraic Geometry Complex Variables The aims of this paper are to answer several conjectures and questions about multiplier spectrum of rational maps and to give new proofs of several rigidity theorems in complex dynamics, by combining tools from complex and non-archimedean dynamics. A remarkable theorem due to McMullen asserts that aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston's rigidity theorem for post-critically finite maps, in which Teichmüller theory is an essential tool. We will give a new proof of McMullen's theorem without using quasiconformal maps or Teichmüller theory. We show that aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen's theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik. We show that a rational map is exceptional if and only if one of the following holds (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field; (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor. |
| title | Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics |
| topic | Dynamical Systems Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2205.13258 |