On the growth rate inequality for self-maps of the sphere

Héctor Barge; Luis Hernández-Corbato

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arXiv · 2301.08543

On the growth rate inequality for self-maps of the sphere

Héctor Barge·Luis Hernández-Corbato

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Abstract

Let $S^m = \{x_0^2 + x_1^2 + \cdots + x_m^2 = 1\}$ and $P = \{x_0 = x_1 = 0\} \cap S^m$. Suppose that $f$ is a self--map of $S^m$ such that $f^{-1}(P) = P$ and $|\mathrm{deg}(f_{|P})| < |\mathrm{deg}(f)|$. Then, the number of fixed points of $f^n$ grows at least exponentially with base $|d| > 1$, where $d = \mathrm{deg}(f)/\mathrm{deg}(f_{|P}) \in \mathbb Z$.

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Héctor Barge, Luis Hernández-Corbato. 2023-01-20. On the growth rate inequality for self-maps of the sphere. https://arxiv.org/abs/2301.08543

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