Reading analytic invariants of parabolic diffeomorphisms from their orbits
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866913970003116032 |
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| author | Klimes, Martin Mardesic, Pavao Radunovic, Goran Resman, Maja |
| author_facet | Klimes, Martin Mardesic, Pavao Radunovic, Goran Resman, Maja |
| contents | In this paper we study germs of diffeomorphisms in the complex plane. We address the following problem: How to read a diffeomorphism $f$ knowing one of its orbits $\mathbb{A}$?
We solve this problem for parabolic germs. This is done by associating to the orbit ${\mathbb{A}}$ a function that we call the dynamic theta function $Θ_{\mathbb{A}}$. We prove that the function $Θ_{\mathbb{A}}$ is $2πi\mathbb{Z}$-resurgent. We show that one can obtain the sectorial Fatou coordinate as a Laplace-type integral transform of the function $Θ_{\mathbb{A}}$. This enables one to read the analytic invariants of a diffeomorphism from the theta function of one of its orbits.
We also define a closely related fractal theta function $\tildeΘ_{\mathbb{A}}$, which is inspired by and generalizes the geometric zeta function of a fractal string, and show that it also encodes the analytic invariants of the diffeomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_14324 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Reading analytic invariants of parabolic diffeomorphisms from their orbits Klimes, Martin Mardesic, Pavao Radunovic, Goran Resman, Maja Dynamical Systems Complex Variables 42A38, 32H50, 37G05 In this paper we study germs of diffeomorphisms in the complex plane. We address the following problem: How to read a diffeomorphism $f$ knowing one of its orbits $\mathbb{A}$? We solve this problem for parabolic germs. This is done by associating to the orbit ${\mathbb{A}}$ a function that we call the dynamic theta function $Θ_{\mathbb{A}}$. We prove that the function $Θ_{\mathbb{A}}$ is $2πi\mathbb{Z}$-resurgent. We show that one can obtain the sectorial Fatou coordinate as a Laplace-type integral transform of the function $Θ_{\mathbb{A}}$. This enables one to read the analytic invariants of a diffeomorphism from the theta function of one of its orbits. We also define a closely related fractal theta function $\tildeΘ_{\mathbb{A}}$, which is inspired by and generalizes the geometric zeta function of a fractal string, and show that it also encodes the analytic invariants of the diffeomorphism. |
| title | Reading analytic invariants of parabolic diffeomorphisms from their orbits |
| topic | Dynamical Systems Complex Variables 42A38, 32H50, 37G05 |
| url | https://arxiv.org/abs/2112.14324 |