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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.19829 |
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| _version_ | 1866916967194034176 |
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| author | He, Jiaxing Hou, Bingzhe Wang, Xiao Xin, Yue |
| author_facet | He, Jiaxing Hou, Bingzhe Wang, Xiao Xin, Yue |
| contents | As well-known, inner functions play an important role in the study of bounded analytic function theory. In recent years, persistence module theory, as a main tool applied to Topological Data Analysis, has received widespread attention. In this paper, we aim to use persistence module theory to study inner functions. We introduce the persistence modules arised from the level sets of inner functions. Some properties of these persistence modules are shown. In particular, we prove that the persistence modules (potentially not of locally finite type) induced by a class of inner functions have interval module decompositions. Furthermore, we demonstrate that the interleaving distance of the persistence modules is continuous with respect to the supremum norm for a class of Blaschke products, which could be used to discuss the path-connectedness of Blaschke products. As an example, we provide an explicit formula for the interleaving distance of the persistence modules induced by the Blaschke products with order two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19829 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Persistence modules induced by inner functions He, Jiaxing Hou, Bingzhe Wang, Xiao Xin, Yue Algebraic Topology Complex Variables 55N31, 30J05, 30J10, 16D70 As well-known, inner functions play an important role in the study of bounded analytic function theory. In recent years, persistence module theory, as a main tool applied to Topological Data Analysis, has received widespread attention. In this paper, we aim to use persistence module theory to study inner functions. We introduce the persistence modules arised from the level sets of inner functions. Some properties of these persistence modules are shown. In particular, we prove that the persistence modules (potentially not of locally finite type) induced by a class of inner functions have interval module decompositions. Furthermore, we demonstrate that the interleaving distance of the persistence modules is continuous with respect to the supremum norm for a class of Blaschke products, which could be used to discuss the path-connectedness of Blaschke products. As an example, we provide an explicit formula for the interleaving distance of the persistence modules induced by the Blaschke products with order two. |
| title | Persistence modules induced by inner functions |
| topic | Algebraic Topology Complex Variables 55N31, 30J05, 30J10, 16D70 |
| url | https://arxiv.org/abs/2509.19829 |