Elliptopia: A Dimension-Indexed Dynamical System for Iterated Pearson Correlation

This technical note introduces the term Elliptopia for the dimension-indexed discrete dynamical system generated by iterating the Pearson correlation map on correlation matrices. For square matrices of dimension two and higher, the ambient geometric space is the corresponding correlation elliptope, while the natural dynamical state space is its Pearson-nondegenerate part. The note proves that the all-ones matrix is the unique Pearson-degenerate correlation matrix in the elliptope and that the Pearson-nondegenerate part of the elliptope is forward invariant under repeated Pearson correlation. Consequently, Pearson correlation defines a genuine self-map of this natural nondegenerate state space in every dimension. Taking the disjoint union of these forward-invariant state spaces across dimensions yields a dimension-preserving discrete dynamical system, denoted Elliptopia. The note also establishes an exact first-step dichotomy for general nondegenerate initial matrices. After one Pearson step, the trajectory either reaches the Pearson-degenerate all-ones matrix, in which case the next Pearson step is undefined, or enters the forward-invariant nondegenerate part of the elliptope and remains there for all subsequent iterates. For independent product-uniform initialization, the exceptional degenerate first image has probability zero in dimension three and higher, yielding almost-sure forward well-posedness. The first Pearson step may therefore be viewed as the transition from a nondegenerate initial matrix to correlation-elliptope dynamics, while Elliptopia describes the subsequent forward-invariant evolution. The framework provides a common mathematical setting for the study of fixed points, relative stability, basin geometry, empirical laws, finite-step probabilistic bounds, and early-trajectory representations arising from iterated Pearson correlation. The present note establishes the natural forward-invariant state space and forward well-posedness of the iteration; questions of global convergence require separate dynamical analysis.