Algebraic dynamics and recursive inequalities
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915218043437056 |
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| author | Xie, Junyi |
| author_facet | Xie, Junyi |
| contents | We get three basic results in algebraic dynamics:
(1). We give the first algorithm to compute the dynamical degrees to arbitrary precision.
(2). We prove that for a family of dominant rational self-maps, the dynamical degrees are lower semi-continuous with respect to the Zariski topology. This implies a conjecture of Call and Silverman.
(3). We prove that the set of periodic points of a cohomologically hyperbolic rational self-map is Zariski dense.
Moreover, we show that, after a large iterate, every degree sequence grows almost at a uniform rate. This property is not satisfied for general submultiplicative sequences. Finally, we prove the Kawaguchi-Silverman conjecture for a class of self-maps of projective surfaces including all the birational ones.
In fact, for every dominant rational self-map, we find a family of recursive inequalities of some dynamically meaningful cycles. Our proofs are based on these inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic dynamics and recursive inequalities Xie, Junyi Dynamical Systems Algebraic Geometry We get three basic results in algebraic dynamics: (1). We give the first algorithm to compute the dynamical degrees to arbitrary precision. (2). We prove that for a family of dominant rational self-maps, the dynamical degrees are lower semi-continuous with respect to the Zariski topology. This implies a conjecture of Call and Silverman. (3). We prove that the set of periodic points of a cohomologically hyperbolic rational self-map is Zariski dense. Moreover, we show that, after a large iterate, every degree sequence grows almost at a uniform rate. This property is not satisfied for general submultiplicative sequences. Finally, we prove the Kawaguchi-Silverman conjecture for a class of self-maps of projective surfaces including all the birational ones. In fact, for every dominant rational self-map, we find a family of recursive inequalities of some dynamically meaningful cycles. Our proofs are based on these inequalities. |
| title | Algebraic dynamics and recursive inequalities |
| topic | Dynamical Systems Algebraic Geometry |
| url | https://arxiv.org/abs/2402.12678 |