[Paper Review] Indices of the iterates of $R^3$-homeomorphisms at Lyapunov stable fixed points

Francisco R. Ruiz del Portal, José Manuel Salazar|ArXiv.org|Apr 18, 2007

Advanced Differential Equations and Dynamical Systems1 references4 citations

TL;DR

This paper constructs orientation-preserving $ℝ^3$-homeomorphisms with a Lyapunov stable fixed point at the origin such that the absolute value of the fixed point index of the $m$-th iterate grows faster than any given positive sequence $\{c_m\}$, proving that the sequence of indices can be unbounded. The construction uses flow boxes and vector field modifications via Wilson's theorem to embed complex dynamics in cones around the origin, demonstrating a fundamental difference from planar dynamics where indices are bounded and periodic.

ABSTRACT

Given any positive sequence (\{c_n\}_{n \in {\Bbb N}}), we construct orientation preserving homeomorphisms (f:{\Bbb R}^3 o {\Bbb R}^3) such that (Fix(f)=Per(f)=\{0\}), (0) is Lyapunov stable and (\limsup \frac{|i(f^m, 0)|}{c_m}= \infty). We will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms.

Motivation & Objective

• To investigate the behavior of fixed point index sequences for iterates of $ℝ^3$-homeomorphisms with Lyapunov stable fixed points.
• To determine whether such index sequences can grow arbitrarily fast relative to any prescribed positive sequence $\{c_m\}$.
• To contrast the dynamics in $ℝ^3$ with the well-understood periodic behavior of indices in the planar case.
• To resolve Problem 2.3.1 from [22] by showing the failure of boundedness in index growth for stable fixed points in $ℝ^3$.

Proposed method

• Construct a vector field $X(x,y,z) = (-x,-y,z)$ on the upper half-space $\pi^+$ with a flow that captures orbits near the origin.
• Apply Wilson’s theorem to modify the vector field in specific flow boxes $U_k$ and $V_k$, creating localized dynamics with controlled periodic orbits.
• Define a smooth, compactly supported vector field $X_1 = \gamma G$ using a bump function $\gamma$ vanishing only at the origin to ensure smoothness and compact support.
• Construct a flow $\psi$ on $\pi^+$ with countably many periodic orbits and choose a decreasing sequence $t_n \to 0$ such that $\psi(t_n, \cdot)$ has only the origin as fixed and periodic point.
• Paste copies of the time-$t_n$ map $\psi(t_n, \cdot)$ into symmetric cones $E(T_{j,m})$ around the origin using cone-preserving homeomorphisms $h_m$, ensuring Lyapunov stability.
• Combine the resulting local dynamics with a global homeomorphism $h_0$ that matches the local maps on each cone and preserves the origin as the only fixed and periodic point.

Experimental results

Research questions

• RQ1 Can the sequence of fixed point indices $|i(f^m, 0)|$ grow faster than any prescribed positive sequence $\{c_m\}$ for an orientation-preserving $ℝ^3$-homeomorphism with a Lyapunov stable fixed point at 0?
• RQ2 Does the failure of boundedness in index growth for stable fixed points in $ℝ^3$ contrast sharply with the bounded and periodic behavior observed in planar homeomorphisms?
• RQ3 Can such unbounded index sequences be constructed even when the maximal invariant set in a neighborhood is not a singleton, but includes a 2-disc and a countable family of circles?
• RQ4 Is the conjecture of Shub and Sullivan, which relates growth of periodic points to Lefschetz numbers, invalidated when replaced by fixed point indices in $ℝ^3$?
• RQ5 Can the index sequence be made unbounded while preserving Lyapunov stability and ensuring that the fixed and periodic point sets are trivial?

Key findings

• For any positive sequence $\{c_m\}$, there exists an orientation-preserving $ℝ^3$-homeomorphism $f$ such that $\limsup_{m \to \infty} \frac{|i(f^m, 0)|}{c_m} = \infty$, proving unbounded growth of indices.
• The constructed homeomorphism $f$ satisfies $Fix(f) = Per(f) = \{0\}$, and $0$ is Lyapunov stable, with $Inv(B, f)$ being the closed 2-disc $B \cap \{z=0\}$ for any closed ball $B$ centered at the origin.
• The homeomorphism $h$ in Theorem 2 satisfies $\limsup_{m \to \infty} \frac{\log |i(h^m, 0)|}{m} = \infty$, indicating super-exponential growth of the index magnitude.
• The sequence of indices for $h$ coincides with that of a related homeomorphism $f$ from Theorem 1, and both are limits of homeomorphisms for which $\{0\}$ is an isolated invariant set.
• The construction shows that the method of prime ends used in planar dynamics fails in $ℝ^3$ because the associated prime ends are not isolated invariant sets when $Inv(B,f)$ contains a 2-disc.
• For orientation-reversing homeomorphisms $S \circ F$ and $S \circ H$, the odd iterates have index 1, while even iterates satisfy $i((S \circ H)^{2k}, 0) = -1 + 2i(h^{2k}, 0)$, reflecting the unbounded growth in the even indices.