Vector Invariance and Structural Closure of Julia-Type Iterations in Clifford Algebra

Abstract

In this paper, we introduce a Clifford algebra framework for Julia-type dynamics driven by the geometric product. The nonlinear iteration \[ f(\vec{x}) = (\vec{x}\diamond \vec{n})^p \diamond \vec{n} + \vec{c}, \qquad p \ge 2, \] is studied in a real $n$-dimensional inner-product space $V$, where $\vec{x}, \vec{n}, \vec{c} \in V$ and $\vec{n}$ is a unit vector. The main result reveals a previously unreported invariance phenomenon: although the geometric product generates higher-grade multivector components at intermediate stages, a built-in grade-reduction mechanism ensures complete collapse back to the vector subspace. Consequently, the Clifford Julia operator is shown to be closed on $V$, and the iteration defines a well-posed nonlinear dynamical system in arbitrary dimensions. This invariance is established through a structural decomposition of the Clifford product and an inductive closure argument, supported by explicit verification in low-dimensional cases and a general proof in $\mathbb{R}^n$. The results demonstrate that classical Julia dynamics can be consistently extended beyond the complex plane into higher-dimensional geometric algebra without loss of geometric interpretability. The framework opens a new direction for fractal-type dynamics in Clifford algebras, providing a unified algebraic setting for higher-dimensional invariant-preserving iterative systems.