PreprintCC BY 4.0arXiv
Dimension of the Green measure of skew products with dominant exponent in the fibers
Abstract
Let f(z,w)=(p(z),q(z,w)) be a polynomial skew product of ℂ2 of degree dgeqslant 2 and let μ and μp be the Green measures associated to f and p respectively. Let λ1 > λ2 and λp be the Lyapunov exponents of (f,μ) and (p,μp) respectively. We show that if λ2 = λp, then μ is exact dimensional and: dimHμ=(log d)/λ1+(log d)/λ2. The proof relies on the continuity of the conditional measures obtained by Jonsson, on the fact that μ has constant Jacobian d2, and on a fibered version of a normal form theorem for the inverse branches of the iterates of f.
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BibTeX
@misc{genes2026dimension,
title = {{Dimension of the Green measure of skew products with dominant exponent in the fibers}},
author = {Clara Genes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19812},
eprint = {2610.19812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19812},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dimension of the Green measure of skew products with dominant exponent in the fibers
AU - Clara Genes
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19812
UR - https://arxiv.org/abs/2610.19812
AB - Let f(z,w)=(p(z),q(z,w)) be a polynomial skew product of ℂ^2 of degree dgeqslant 2 and let μ and μ_p be the Green measures associated to f and p respectively. Let λ_1 > λ_2 and λ_p be the Lyapunov exponents of (f,μ) and (p,μ_p) respectively. We show that if λ_2 = λ_p, then μ is exact dimensional and: dim_Hμ=(log d)/λ_1+(log d)/λ_2. The proof relies on the continuity of the conditional measures obtained by Jonsson, on the fact that μ has constant Jacobian d^2, and on a fibered version of a normal form theorem for the inverse branches of the iterates of f.
ER -