Fixed points in non-invariant plane continua

TL;DR

This paper extends fixed point theorems from real intervals to complex continua, including dendrites and holomorphic maps, identifying conditions under which fixed points must exist or rotation occurs.

Contribution

It generalizes fixed point results to complex continua like dendrites and positively oriented maps, including non-invariant cases, and establishes fixed point existence for holomorphic maps.

Findings

Fixed points exist in certain complex continua under specified conditions.

Holomorphic maps must have fixed points or exhibit rotation within the continuum.

Extension of classical fixed point theorems to complex and dendritic structures.

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 $a$.