Approximation of non-Archimedean Lyapunov exponents and applications over global fields. (English) Zbl 1459.37098
Summary: Let \(K\) be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value. We establish an approximation of the Lyapunov exponent of a rational map \(f\) of \(\mathbb{P}^1\) of degree \(d>1\) defined over \(K\) in terms of the multipliers of periodic points of \(f\) having the formally exact period \(n\), with an explicit error estimate in terms of \(f,n\), and \(d\). As an immediate consequence, we obtain an estimate on the blow-up of the Lyapunov exponent function near a pole in one-dimensional parameter families of rational maps over \(K\).
Combined with an improvement of our former archimedean counterpart, this non-archimedean quantitative approximation of Lyapunov exponents allows us to establish
Combined with an improvement of our former archimedean counterpart, this non-archimedean quantitative approximation of Lyapunov exponents allows us to establish
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- a quantification of Silverman’s and Ingram’s recent comparison between the critical height and any ample height on the dynamical moduli space \(\mathcal{M}_d(\overline{\mathbb{Q}})\) except for the flexible Lattès locus,
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- an improvement of McMullen’s finiteness of the multiplier maps in two aspects: reduction to multipliers of cycles having a given formally exact period, and
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- an explicit computation on the magnitude of the formally exact period of cycles, and a characterization of non-affine isotrivial rational maps defined over a function field \(\mathbb{C}(X)\) of a complex normal projective variety \(X\) in terms of the growth of the degree of the multipliers of cycles.
MSC:
| 37P30 | Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems |
| 37P45 | Families and moduli spaces in arithmetic and non-Archimedean dynamical systems |
| 37P05 | Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps |
| 11S82 | Non-Archimedean dynamical systems |
| 26E30 | Non-Archimedean analysis |
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