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Absolute winning properties of exceptional sets for rational maps
Abstract
We study Schmidt's game on the Julia set J(f) of weak hyperbolic rational maps f:CC→CC. For each y∈ J(f), let E(f,y) be the set of points in J(f) whose forward orbits stay a positive distance from y. We prove that E(f,y) is absolute winning on J(f) for each y ∈ J(f). To do this, we first prove the absolute winning property for the exceptional set associated to the critical points by a potential game argument, and then use uniform expansion to obtain the result for each target point y.
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BibTeX
@misc{ji2026absolute,
title = {{Absolute winning properties of exceptional sets for rational maps}},
author = {Haoyang Ji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27796},
eprint = {2610.27796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27796},
}
RIS
TY - UNPB
TI - Absolute winning properties of exceptional sets for rational maps
AU - Haoyang Ji
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27796
UR - https://arxiv.org/abs/2610.27796
AB - We study Schmidt's game on the Julia set J(f) of weak hyperbolic rational maps f:CC→CC. For each y∈ J(f), let E(f,y) be the set of points in J(f) whose forward orbits stay a positive distance from y. We prove that E(f,y) is absolute winning on J(f) for each y ∈ J(f). To do this, we first prove the absolute winning property for the exceptional set associated to the critical points by a potential game argument, and then use uniform expansion to obtain the result for each target point y.
ER -