# Quantitative logarithmic equidistribution of the crucial measures

Quantitative logarithmic equidistribution of the crucial measures Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let $$\phi \in K(z)$$ ϕ ∈ K ( z ) with $$\deg (\phi )\ge 2$$ deg ( ϕ ) ≥ 2 . Recently, Rumely introduced a family of discrete probability measures $$\{\nu _{\phi ^n}\}$$ { ν ϕ n } on the Berkovich line $$\mathbf{P }^1_{\text {K}}$$ P K 1 over K which carry information about the reduction of conjugates of $$\phi$$ ϕ . In a previous article, the author showed that the measures $$\nu _{\phi ^n}$$ ν ϕ n converge weakly to the canonical measure $$\mu _\phi$$ μ ϕ . In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of $$\mathbf{P }^1_{\text {K}}$$ P K 1 . These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to $$\nu _{\phi ^n}$$ ν ϕ n converge to the potential function attached to $$\mu _\phi$$ μ ϕ , as well as an approximation result for the Lyapunov exponent of $$\phi$$ ϕ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in Number Theory Springer Journals

# Quantitative logarithmic equidistribution of the crucial measures

, Volume 4 (1) – Feb 8, 2018
32 pages

/lp/springer_journal/quantitative-logarithmic-equidistribution-of-the-crucial-measures-dpNsLtmdXL
Publisher
Springer International Publishing
Subject
Mathematics; Number Theory
eISSN
2363-9555
D.O.I.
10.1007/s40993-018-0094-1
Publisher site
See Article on Publisher Site

### Abstract

Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let $$\phi \in K(z)$$ ϕ ∈ K ( z ) with $$\deg (\phi )\ge 2$$ deg ( ϕ ) ≥ 2 . Recently, Rumely introduced a family of discrete probability measures $$\{\nu _{\phi ^n}\}$$ { ν ϕ n } on the Berkovich line $$\mathbf{P }^1_{\text {K}}$$ P K 1 over K which carry information about the reduction of conjugates of $$\phi$$ ϕ . In a previous article, the author showed that the measures $$\nu _{\phi ^n}$$ ν ϕ n converge weakly to the canonical measure $$\mu _\phi$$ μ ϕ . In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of $$\mathbf{P }^1_{\text {K}}$$ P K 1 . These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to $$\nu _{\phi ^n}$$ ν ϕ n converge to the potential function attached to $$\mu _\phi$$ μ ϕ , as well as an approximation result for the Lyapunov exponent of $$\phi$$ ϕ .

### Journal

Research in Number TheorySpringer Journals

Published: Feb 8, 2018

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