AN EQUIVARIANT TOPOLOGICAL INVARIANT FOR GRADIENT FLOWS OF HARMONIC POTENTIALS IN THE PLANE

The distribution of zeros of entire functions of finite order has been a central theme in complex analysis, particularly when such functions are subject to functional equations that impose reflection symmetries. This article introduces a new topological invariant for a class of entire functions of order at most 1 that admit an involutive functional symmetry of the form f(s) = ω(s)f(1 − s), with ω holomorphic and |ω(s)| = 1 for all s. Associated with each such function is the logarithmic potential u(s) = log |f(s)|, harmonic outside the zeros. The gradient field X = ∇u generates a flow, and the symmetry induces an anti-equivariant action of Z2 on this flow. A key consequence is that the critical line L = {ℜs = 1/2} serves as a rigid barrier: the flow is tangent to L outside the zeros, dividing the plane into two invariant half-planes. Using the equivariant Conley index with respect to this Z2 action, we construct a global topological invariant. Zeros outside the critical line produce free orbits with non-trivial equivariant index, incompatible with the asymptotic rigidity at infinity imposed by the controlled growth. The main theorem establishes a confinement principle for the class, revealing a novel interaction between harmonic analysis, gradient flows, and equivariant dynamical topology.