[Paper Review] Connected components of partition preserving diffeomorphisms

Sergiy Maksymenko|arXiv (Cornell University)|Jun 1, 2008

Mathematical Dynamics and Fractals7 references3 citations

TL;DR

This paper characterizes the connected components of diffeomorphisms preserving a homogeneous polynomial $f:\mathbb{R}^2 \to \mathbb{R}$, showing that the identity components $\mathcal{S}_{\mathrm{id}}(f)^r$ for $r \geq 1$ coincide, while $\mathcal{S}_{\mathrm{id}}(f)^1 \neq \mathcal{S}_{\mathrm{id}}(f)^0$ if and only if $f$ is a product of at least two distinct definite quadratic forms. The result relies on analyzing singular partitions and isotopies preserving level sets of $f$. The key contribution is a complete classification of when higher differentiability classes yield strictly smaller identity components in the stabilizer group of $f$.

ABSTRACT

Let $f:\mathbb{R}^2 o \mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 o \mathbb{R}^2$ preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq \infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $S(f,\infty) = \cdots = S(f,1)$ for all such $f$ and that $S(f,1) ot= S(f,0)$ if and only if $f$ is a product of at least two distinct irreducible over $\mathbb{R}$ quadratic forms.

Motivation & Objective

• Understand the structure of the identity component $\mathcal{S}_{\mathrm{id}}(f)^r$ of the group of diffeomorphisms preserving a homogeneous polynomial $f:\mathbb{R}^2 \to \mathbb{R}$, under the weak Whitney $C^r_W$-topology.
• Clarify the relationship between the identity components $\mathcal{S}_{\mathrm{id}}(f)^r$ for different $r$, particularly when they stabilize or differ.
• Determine the precise condition under which $\mathcal{S}_{\mathrm{id}}(f)^1 \neq \mathcal{S}_{\mathrm{id}}(f)^0$, which corresponds to a topological obstruction in the diffeomorphism group.
• Investigate how the algebraic structure of $f$, especially its factorization into linear and definite quadratic forms, affects the connectedness of the stabilizer group.
• Establish a general framework for partition-preserving diffeomorphisms via singular partitions and invariant contractions, applicable to broader problems in singularity theory.

Proposed method

• The paper defines $r$-homotopies as continuous families of $C^r$-maps $H: \mathbb{R}^2 \times I \to \mathbb{R}^2$ whose partial derivatives up to order $r$ depend continuously on $(x,t)$, which correspond to continuous paths in $C^r_W$-topology.
• By analyzing the singular partition $\Theta_G$ induced by the level sets of $f$, the paper relates the stabilizer group $\mathcal{S}(f)$ to the group $\mathcal{D}(\Theta_G)$ of diffeomorphisms preserving this partition.
• Using the local flow $\Phi$ generated by a vector field $G$, the paper constructs a shift map $\varphi$ whose image captures diffeomorphisms isotopic to identity through $1$-homotopies.
• The proof relies on showing that for $f$ a product of at least two distinct definite quadratic forms, the tangent map at the origin of any $h \in \mathcal{S}_{\mathrm{id}}(f)^1$ must be the identity, while $\mathcal{S}_{\mathrm{id}}(f)^0$ includes $-\mathrm{id}$, leading to a strict inclusion.
• Key results from prior works, such as $\mathcal{S}_{\mathrm{id}}(f)^\infty = \mathcal{S}_{\mathrm{id}}(f)^1$ and the discreteness of linear symmetries $\mathcal{LS}(f)$ in certain cases, are applied to establish the chain of equalities and strict inequalities.
• The classification of $f$ into cases (A)–(E) based on degree and factorization allows case-by-case analysis of the stabilizer group's identity components.

Experimental results

Research questions

• RQ1 Under what conditions does $\mathcal{S}_{\mathrm{id}}(f)^1$ strictly differ from $\mathcal{S}_{\mathrm{id}}(f)^0$?
• RQ2 How do the identity components $\mathcal{S}_{\mathrm{id}}(f)^r$ behave across different differentiability classes $r$ for a homogeneous polynomial $f:\mathbb{R}^2 \to \mathbb{R}$?
• RQ3 What role does the factorization of $f$ into irreducible real factors—especially definite quadratic forms—play in determining the topology of the stabilizer group?
• RQ4 Can the identity component of the stabilizer group be characterized via flows or isotopies that preserve the level sets of $f$?
• RQ5 How does the tangent map at the origin constrain the possible elements of $\mathcal{S}_{\mathrm{id}}(f)^1$ compared to $\mathcal{S}_{\mathrm{id}}(f)^0$?

Key findings

• The identity components $\mathcal{S}_{\mathrm{id}}(f)^r$ for $r \geq 1$ are all equal: $\mathcal{S}_{\mathrm{id}}(f)^\infty = \cdots = \mathcal{S}_{\mathrm{id}}(f)^1$ for every homogeneous polynomial $f:\mathbb{R}^2 \to \mathbb{R}$.
• An element $h \in \mathcal{S}_{\mathrm{id}}(f)^1$ must satisfy $T_0 h = \mathrm{id}_{\mathbb{R}^2}$, meaning its derivative at the origin is trivial.
• Whenever $f$ is a product of at least two distinct definite quadratic forms, $\mathcal{S}_{\mathrm{id}}(f)^1 \neq \mathcal{S}_{\mathrm{id}}(f)^0$, because $-\mathrm{id}_{\mathbb{R}^2} \in \mathcal{S}_{\mathrm{id}}(f)^0$ but $-\mathrm{id}_{\mathbb{R}^2} \notin \mathcal{S}_{\mathrm{id}}(f)^1$.
• In the case where $f$ is a product of distinct definite quadratic forms, $\mathcal{S}_{\mathrm{id}}(f)^0 = \mathcal{S}^+(f)$, the orientation-preserving component of the stabilizer.
• When $f$ is a product of at least two distinct definite quadratic forms, $\mathcal{S}_{\mathrm{id}}(f)^1$ is a proper subgroup of $\mathcal{S}_{\mathrm{id}}(f)^0$, indicating a topological obstruction to isotopy through $C^1$-smooth maps.
• For all other $f$, including those with linear or non-definite quadratic factors, $\mathcal{S}_{\mathrm{id}}(f)^1 = \mathcal{S}_{\mathrm{id}}(f)^0$, so the identity components stabilize at $r=1$.