Fixed points in non-invariant plane continua

Abstract

If $f:[a,b]\to \mathbb{R}$, with $a<b$, is continuous and such that $a$ and $b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in $I$. Suppose that $f:\mathbb{C}\to \mathbb{C}$ is map and $X$ is a continuum. We extend the above for certain continuous maps of dendrites $X\to D,X\subset D$ and for positively oriented maps $f:X\to \mathbb{C},X\subset \mathbb{C}$ with the continuum $X$ not necessarily invariant. Then we show that in certain cases a holomorphic map $f:\mathbb{C}\to \mathbb{C}$ must have a fixed point $a$ in a continuum $X$ so that either $a\in \mathrm{Int}(X)$ or $f$ exhibits rotation at $a$.

21 pages with corrected references

References in corpus (1)

• The plane fixed point problem

Cited by in corpus (3)

• The plane fixed point problem
• Locally connected models for Julia sets
• The Julia sets of basic uniCremer polynomials of arbitrary degree