Bi-Hölder equivalence of real analytic functions. (English) Zbl 1543.14004
From a historical context, the authors present the following question:
Question 1. Is there a family of polynomials \(\{f_t\}_{t \in I}\) (where \(I\) is a real interval) with uniformly bounded degree such that \(f_t\) is not Hölder equivalent to\(f_s\) for any \(s \neq t\) ?
In addition to preparing the demonstration of the positive answer to this question, it is demonstrated that the Hölder equivalence of function germs \((\mathbb{R}^2, 0) \to (\mathbb{R}, 0) \) admits continuous moduli.
In particular, they approach an example in a very interesting way, namely, an adaptation of the example built by A. Bodin [Math. Nachr. 294, No. 7, 1295–1310 (2021; Zbl 1531.32004)] to show the existence of continuous moduli for the bi-Lipschitz classification of two real variable functions at infinity.
Above all, they address elements regarding the study of singularities up to Hölder homeomorphisms was already considered in the paper [A. Fernandes et al., Bull. Lond. Math. Soc. 50, No. 5, 874–886 (2018; Zbl 1405.14005)] as well as germs of analytic functions \(f: (\mathbb{R}^2, 0) \to (\mathbb{R}, 0) \) up to change of coordinates which are Hölder regular and their inverse are also Hölder regular.
Question 1. Is there a family of polynomials \(\{f_t\}_{t \in I}\) (where \(I\) is a real interval) with uniformly bounded degree such that \(f_t\) is not Hölder equivalent to\(f_s\) for any \(s \neq t\) ?
In addition to preparing the demonstration of the positive answer to this question, it is demonstrated that the Hölder equivalence of function germs \((\mathbb{R}^2, 0) \to (\mathbb{R}, 0) \) admits continuous moduli.
In particular, they approach an example in a very interesting way, namely, an adaptation of the example built by A. Bodin [Math. Nachr. 294, No. 7, 1295–1310 (2021; Zbl 1531.32004)] to show the existence of continuous moduli for the bi-Lipschitz classification of two real variable functions at infinity.
Above all, they address elements regarding the study of singularities up to Hölder homeomorphisms was already considered in the paper [A. Fernandes et al., Bull. Lond. Math. Soc. 50, No. 5, 874–886 (2018; Zbl 1405.14005)] as well as germs of analytic functions \(f: (\mathbb{R}^2, 0) \to (\mathbb{R}, 0) \) up to change of coordinates which are Hölder regular and their inverse are also Hölder regular.
Reviewer: Eliris Cristina Rizziolli (Rio Claro)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 32S50 | Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants |
References:
| [1] | Bodin, A., Bilipschitz equivalence of polynomials, Math. Nachr., 294, 7, 1295-1310, 2021 · Zbl 1531.32004 · doi:10.1002/mana.201900152 |
| [2] | Fernandes, A.; Sampaio, JE; Silva, JP, Hölder equivalence of complex analytic curve singularities, Bull. London Math. Soc., 50, 5, 874-886, 2018 · Zbl 1405.14005 · doi:10.1112/blms.12192 |
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| [7] | Kuo, TC, On classification of real singularities, Invent. Math., 82, 257-262, 1985 · Zbl 0587.32018 · doi:10.1007/BF01388802 |
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