# H\"older equivalence of complex analytic curve singularities

## TL;DR

This paper establishes that the sequence of characteristic exponents and intersection multiplicities are invariants under Lipschitz (specifically H"older) homeomorphisms for complex analytic curve germs, highlighting their geometric significance.

## Contribution

It proves that differences in characteristic exponents prevent H"older homeomorphism and characterizes invariants for complex plane curve germs under such mappings.

## Key findings

- Characteristic exponents are Lipschitz invariants.
- Different characteristic exponents imply no $eta$-H"older homeomorphism.
- Intersection multiplicities are preserved under H"older homeomorphisms.

## Abstract

We prove that if two germs of irreducible complex analytic curves at $0\in\mathbb{C}^2$ have different sequence of characteristic exponents, then there exists $0<\alpha<1$ such that those germs are not $\alpha$-H\"older homeomorphic. For germs of complex analytic plane curves with several irreducible components we prove that if any two of them are $\alpha$-H\"older homeomorphic, for all $0<\alpha<1$, then there is a correspondence between their branches preserving sequence of characteristic exponents and intersection multiplicity of pair of branches. In particular, we recovery the sequence of characteristic exponents of the branches and intersection multiplicity of pair of branches are Lipschitz invariant of germs of complex analytic plane curves.