Real non-attractive fixed point conjecture for complex harmonic functions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916861667442688 |
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| author | Vaseem, Mohd |
| author_facet | Vaseem, Mohd |
| contents | We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function $f=h+\overline{g}$ is polynomial (rational) if both $h$ and $g$ are polynomials (rational functions) of degree at least 2. We show that every such function with a super-attracting fixed point has a $\mathfrak{h}$-fixed point $ζ=μ+\overlineω$ such that the real parts of its multipliers satisfy $\text{Re}(\partial_z h(μ)) \geq 1$ and $\text{Re}(\partial_z g(ω)) \geq 1$. For polynomial harmonic functions, this holds even without super-attracting conditions. We provide explicit examples, visualizations, and discuss problem for transcendental harmonic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Real non-attractive fixed point conjecture for complex harmonic functions Vaseem, Mohd Complex Variables We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function $f=h+\overline{g}$ is polynomial (rational) if both $h$ and $g$ are polynomials (rational functions) of degree at least 2. We show that every such function with a super-attracting fixed point has a $\mathfrak{h}$-fixed point $ζ=μ+\overlineω$ such that the real parts of its multipliers satisfy $\text{Re}(\partial_z h(μ)) \geq 1$ and $\text{Re}(\partial_z g(ω)) \geq 1$. For polynomial harmonic functions, this holds even without super-attracting conditions. We provide explicit examples, visualizations, and discuss problem for transcendental harmonic functions. |
| title | Real non-attractive fixed point conjecture for complex harmonic functions |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2507.18414 |