Le Roux Index in Planar Dynamics

• The Le Roux index is a topological invariant for Brouwer homeomorphisms that measures orbit winding through continuous angle-of-motion calculations.
• It generalizes the classical Poincaré–Hopf index by quantifying dynamical relationships via displacement vector fields and transverse foliation methods.
• Taking half-integer values, the invariant reveals Reeb-type dynamical obstructions and unifies index theory for planar dynamical systems.

The Le Roux index is a topological invariant associated with pairs of orbits of Brouwer homeomorphisms—orientation-preserving, fixed-point-free homeomorphisms of the plane. Serving as a generalization of the classical Poincaré-Hopf index for non-singular planar flows, the Le Roux index quantifies the dynamical relation between two orbits in terms of winding and transverse foliation structure. Its formulation, rooted in conjugacy classes and the geometry of displacement vector fields, enables broad applicability beyond flow maps and provides a unifying framework for understanding index theory within planar dynamical systems (Schuback, 22 Jan 2026).

1. Definition and Construction

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be a Brouwer homeomorphism and $O_1, O_2$ two of its orbits. The associated continuous displacement vector field is $X_f(x)=f(x)-x$. The core construction involves the set $\mathrm{Handel}(f,O_1,O_2)$ of orientation-preserving homeomorphisms $h$ such that $h(O_i)=\mathbb{R}\times\{i\}$ and the isotopy class of $h\circ f\circ h^{-1}$ relative to $\mathbb{R}\times\{1,2\}$ matches one of four canonical models: $$[T,\{1,2\}],\quad [T^{-1},\{1,2\}],\quad [R,\{1,2\}],\quad [R^{-1},\{1,2\}]$$ where $T(x,y)=(x+1,y)$ is the unit translation and $R$ the time-one map of the standard Reeb flow.

Given any $h \in \mathrm{Handel}(f,O_1,O_2)$, define $F = h \circ f \circ h^{-1}$ such that $F(\mathbb{R}\times\{i\}) = \mathbb{R}\times\{i\}$, and $X_F(x,y) = F(x,y)-(x,y)$ is horizontal along both lines. For any path $\alpha:[0,1]\to\mathbb{R}^2$ connecting $\alpha(0)\in\mathbb{R}\times\{1\}$ to $\alpha(1)\in\mathbb{R}\times\{2\}$, define the angle-of-motion map

$s(t) = \Arg(X_F(\alpha(t))) \in [0,2\pi),$

with continuous lift $\widetilde s:[0,1]\to\mathbb{R}$, and $\widetilde s(1)-\widetilde s(0)\in 2\pi\mathbb{Z}$. Since $X_F$ is horizontal at endpoints, $\widetilde s(1)-\widetilde s(0)$ is an integer multiple of $\pi$. The index is then defined by

$$\mathrm{Ind}(f,O_1,O_2) := \frac{1}{2\pi}(\widetilde s(1) - \widetilde s(0)) \in \tfrac12 \mathbb{Z},$$

which is independent of choices of $h$ and $\alpha$. Alternatively, the index admits a “topological angle” formulation using a map $\Theta$ lifted to an appropriate graph, leading to the equivalent expression

$$\mathrm{Ind}(f,O_1,O_2) = \frac{1}{4}\bigl(\Theta(p_2,F(p_2)) - \Theta(p_1,F(p_1))\bigr),$$

for points $p_i\in\mathbb{R}\times\{i\}$.

2. Connection to Transverse Foliations and Le Calvez’s Theory

Let $\mathcal{F}$ be an oriented topological foliation of $\mathbb{R}^2$ such that every leaf is a Brouwer line for $f$; such foliations exist by Le Calvez’s theorem. Given orbits $O_1, O_2$ and properly embedded transverse trajectories $\Gamma_1, \Gamma_2$, a foliation-index $$\mathrm{Ind}(\mathcal{F},\Gamma_1,\Gamma_2)\in\tfrac12\mathbb{Z}$$ is defined analogously, by mapping $\Gamma_i$ to horizontal lines and measuring the winding of a local “leaf-push” vector.

The central result, Theorem A of (Schuback, 22 Jan 2026), asserts: $$\mathrm{Ind}(\mathcal{F},\Gamma_1,\Gamma_2) = \mathrm{Ind}(f,O_1,O_2).$$ This establishes that the Le Roux index is equivalently computable as a foliation angle for any transverse foliation. The proof leverages topological connectivity and unique lifting properties of the angle map, and demonstrates that index computations along the foliation and along the homeomorphism’s graph coincide up to a combinatorial correction which cancels in the final computation.

3. Relation to the Poincaré-Hopf Index

For the particular case where $f = \Phi^1$ is the time-one map of a smooth, non-singular flow $\{\Phi^t\}$ with integral-curve foliation $\mathcal{F}$, the Le Roux index coincides with the classical Poincaré–Hopf winding index: $\mathrm{Ind}(\Phi^1,O_1,O_2) = \frac{1}{2\pi}(\widetilde\Arg X(\alpha(1)) - \widetilde\Arg X(\alpha(0))) = \mathrm{Ind}_{\mathrm{PH}}(\mathcal{F},\phi_1,\phi_2),$ where $\phi_1, \phi_2$ are the integral curves through $O_1, O_2$ respectively. Thus, the modern definition generalizes the classic index for all Brouwer homeomorphisms and recovers traditional results for non-singular flows.

4. Illustrative Examples

Two canonical examples clarify the computation and interpretation of the Le Roux index:

• Horizontal Translation: For $T(x,y) = (x+1, y)$ and orbits $O_1 = \{y=0\}, O_2 = \{y=1\}$, with the identity conjugacy $h = \mathrm{Id}$, $X_F(x,y) = (1, 0)$ is constant and horizontal. A path $\alpha(t) = (0, t)$ results in

$s(t) = \Arg(1, 0) = 0,\qquad \widetilde s(1)-\widetilde s(0) = 0,\qquad \mathrm{Ind}(T, O_1, O_2) = 0.$

No winding is observed.

• Standard Reeb Homeomorphism: For $R$ the time-one map of the Reeb flow, appropriate choices yield two orbits straddling the Reeb-cylinder, with

$\mathrm{Ind}(R, O_1, O_2) = \pm \tfrac12,$

signifying one half-turn in the displacement vector.

5. Invariants, Applications, and Consequences

The alternative foliation-based construction resolves Le Roux’s question by formulating the index intrinsically, independent of homotopy-theoretic conjugacies. The index is a topological (half-integer) invariant of the dynamical relationship between two orbits for any transverse foliation, not restricted to particular representatives.

A nonzero Le Roux index implies the existence of “Reeb-component-type” dynamical obstructions, directly analogizing classical Poincaré–Hopf obstructions to global triviality in planar flows. Beyond pairs of orbits, the index framework enables further “linking-number” invariants for larger sets of orbits and influences patterns and restrictions on permissible transverse foliations (Schuback, 22 Jan 2026).

6. Synthesis and Theoretical Significance

The Schuback formulation of the Le Roux index provides an angle-based, foliation-theoretic perspective that unifies index theory for planar homeomorphisms and flows. It proves that the foliation index is always equal to the canonical Le Roux index for any pair of orbits under any transverse foliation, thereby promoting the index to a robust invariant for planar dynamical systems. This connection not only offers finer insight into the topology of orbit relations but also seeds further developments in dynamical invariants and foliation theory.