TY - JOUR
T1 - Quantitative logarithmic equidistribution of the crucial measures
AU - Jacobs, Kenneth S.
N1 - Funding Information:
The author would like to thank Robert Rumely for helpful conversations in the preparation of this article, as well as the anonymous referees for their useful feedback. The author was partially supported by a Research Training Grant DMS-1344994 of the RTG in Algebra, Algebraic Geometry, and Number Theory, at the University of Georgia. Not applicable. Not applicable. The authors declare that they have no competing interests. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Publisher Copyright:
© 2018, SpringerNature.
PY - 2018/3/1
Y1 - 2018/3/1
N2 - Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let ϕ∈ K(z) with deg (ϕ) ≥ 2. Recently, Rumely introduced a family of discrete probability measures {νϕn} on the Berkovich line PK1 over K which carry information about the reduction of conjugates of ϕ. In a previous article, the author showed that the measures νϕn converge weakly to the canonical measure μϕ. In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of PK1. These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to νϕn converge to the potential function attached to μϕ, as well as an approximation result for the Lyapunov exponent of ϕ.
AB - Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let ϕ∈ K(z) with deg (ϕ) ≥ 2. Recently, Rumely introduced a family of discrete probability measures {νϕn} on the Berkovich line PK1 over K which carry information about the reduction of conjugates of ϕ. In a previous article, the author showed that the measures νϕn converge weakly to the canonical measure μϕ. In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of PK1. These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to νϕn converge to the potential function attached to μϕ, as well as an approximation result for the Lyapunov exponent of ϕ.
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U2 - 10.1007/s40993-018-0094-1
DO - 10.1007/s40993-018-0094-1
M3 - Article
AN - SCOPUS:85041748894
SN - 2363-9555
VL - 4
JO - Research in Number Theory
JF - Research in Number Theory
IS - 1
M1 - 10
ER -