Fixed point indices of iterates of orientation-reversing homeomorphisms

TL;DR

The paper resolves the realizability problem for fixed point index sequences of iterates of orientation-reversing homeomorphisms in $\mathbb{R}^{m}$ with $m\ge 3$ by showing that any sequence of integers satisfying Dold's congruences can be realized as $\mathrm{ind}(f^{n},0)$. It develops a constructive framework in which boundary-preserving maps on $\mathbb{R}^{m}_{+}$ are used to encode Dold coefficients via conical coordinates and Mayer–Vietoris calculations, yielding the coordinate decomposition ${\rm ind}(f^{n},0)=\sum_k a_k {\rm reg}_{k}(n)$ and ${\rm ind}(\bar f^{n},0)= {\rm reg}_{1}(n)+\bar a_d {\rm reg}_{d}(n)$. The second main contribution extends the realizability to orientation-reversing maps without the isolation assumption (in particular in $\mathbb{R}^{3}$), establishing that for any Dold coefficients there exists an orientation-reversing $f$ with $\mathrm{ind}(f^{n},0)=\sum_k a_k {\rm reg}_{k}(n)$; the construction uses a composition of an orientation-preserving core with a reflection and a detailed index analysis via Mayer–Vietoris and Conley-type arguments. Together, these results broaden the understanding of how fixed point index sequences can arise in high-dimensional topological dynamics and have implications for studying periodic solutions through Poincaré maps.

Abstract

We show that any sequence of integers satisfying necessary Dold's congruences is realized as the sequence of fixed point indices of the iterates of an orientation-reversing homeomorphism of $\mathbb{R}^{m}$ for $m\geq 3$. As an element of the construction of the above homeomorphism, we consider the class of boundary-preserving homeomorphisms of $\mathbb{R}^{m}_{+}$ and give the answer to [Problem 10.2, Topol. Methods Nonlinear Anal. 50 (2017), 643 - 667] providing a complete description of the forms of fixed point indices for this class of maps.

Figures (13)

• Figure 1: Examples of various types of phase portraits of $H_{p}$.
• Figure 2: The main and the intermediate conical surfaces in $\mathbb{R}^{3}_{+}$.
• Figure 3: The graphs of maps $\tau$, $\sigma$ and $\mu$.
• Figure 4: The example of graph of $\psi$.
• Figure 5: The splitting of the sector $\mathcal{S}$.

Theorems & Definitions (35)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4
• Corollary 2.5
• Theorem 3.2
• Theorem 3.3
• Remark 3.4
• Lemma 3.5
• proof