Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912076319948800 |
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| author | Hemmingsson, Nils |
| author_facet | Hemmingsson, Nils |
| contents | In this paper, we analyze a certain family of holomorphic correspondences on $\hat{\mathbb C}\times\hat{\mathbb C}$ and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map $F$ is such that for any $a\in \mathbb C$, we have that $(F^n)_*(δ_a)$ converges to a probability measure $μ_F$ for which $F_*(μ_F)=μ_F d$ where $d$ is the degree of $F$. This result is used to show that the minimal Hutchinson invariant set in degree $n$ of a large class of operators and for sufficiently large $n$ exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, this set is a Cantor set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13959 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets Hemmingsson, Nils Dynamical Systems In this paper, we analyze a certain family of holomorphic correspondences on $\hat{\mathbb C}\times\hat{\mathbb C}$ and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map $F$ is such that for any $a\in \mathbb C$, we have that $(F^n)_*(δ_a)$ converges to a probability measure $μ_F$ for which $F_*(μ_F)=μ_F d$ where $d$ is the degree of $F$. This result is used to show that the minimal Hutchinson invariant set in degree $n$ of a large class of operators and for sufficiently large $n$ exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, this set is a Cantor set. |
| title | Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2305.13959 |