Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910554536280064 |
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| author | Silverman, Joseph H. |
| author_facet | Silverman, Joseph H. |
| contents | In this note we give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system f:X-->X, including the dynamical degree d(f), which gives a coarse measure of the geometric complexity of the iterates of f, the arithmetic degree a(f,P), which gives a coarse measure of the arithmetic complexity of the orbit of a an algebraic point P in X, and various versions of the canonical height h_f(P) that provide more refined measures of arithmetic complexity. Emphasis is placed on open problems and directions for further exploration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions Silverman, Joseph H. Number Theory Dynamical Systems Primary: 37P05, Secondary: 37P15, 37P30, 37P55 In this note we give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system f:X-->X, including the dynamical degree d(f), which gives a coarse measure of the geometric complexity of the iterates of f, the arithmetic degree a(f,P), which gives a coarse measure of the arithmetic complexity of the orbit of a an algebraic point P in X, and various versions of the canonical height h_f(P) that provide more refined measures of arithmetic complexity. Emphasis is placed on open problems and directions for further exploration. |
| title | Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions |
| topic | Number Theory Dynamical Systems Primary: 37P05, Secondary: 37P15, 37P30, 37P55 |
| url | https://arxiv.org/abs/2408.01559 |