{"title":"Meromorphic functions whose action on their Julia sets is Non-Ergodic","authors":"Tao Chen, Yunping Jiang, Linda Keen","doi":"arxiv-2409.12127","DOIUrl":null,"url":null,"abstract":"the action of the function on its Julia set is still ergodic if some, but not\nall of the asymptotic values land on infinity, and the remaining ones land on a\ncompact repeller. In this paper, we complete the characterization of ergodicity\nfor Nevanlinna functions but proving that if all the asymptotic values land on\ninfinity, then the Julia set is the whole sphere and the action of the map\nthere is non-ergodic.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Dynamical Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.12127","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
the action of the function on its Julia set is still ergodic if some, but not
all of the asymptotic values land on infinity, and the remaining ones land on a
compact repeller. In this paper, we complete the characterization of ergodicity
for Nevanlinna functions but proving that if all the asymptotic values land on
infinity, then the Julia set is the whole sphere and the action of the map
there is non-ergodic.
如果部分渐近值落在无穷大上，而不是所有渐近值都落在无穷大上，并且剩余的渐近值落在一个紧密排斥者上，那么函数在其 Julia 集上的作用仍然是遍历性的。在本文中，我们完成了对 Nevanlinna 函数遍历性的表征，但证明了如果所有渐近值都落在无穷大上，那么 Julia 集就是整个球面，并且该球面的作用是非遍历性的。
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