Analytic patch trees: branch interface inheritance and fractal dimension fields
For researchers in fractal geometry and complex systems, this work provides a theoretical framework for analytic surface patch trees, though it is an incremental extension of prior curve tree theory.
The paper extends analytic fractal curve trees to surface patch trees, introducing interface curves that transmit analytical states between parent and child patches. It establishes conditions for integrability, well-posedness, and conformality, and shows that patch trees have a natural foliation into curve trees with a smooth Hausdorff dimension field.
The extension of the analytic fractal curve trees of (2601.17490} to analytic surface patch trees reveals a new geometric structure: branch points are replaced by interface curves that transmit the full analytical state of parent patches to their children. These interfaces prove to be central in determining the topology of the surface patch trees, including for the conditions for self-similarity of the interfaces, the patches and thus the trees. We establish the analytic conditions for the integrability and well-posedness of the surface patch trees and introduce further restrictions for conformality. We demonstrate that patch trees have a natural foliation that slices the trees into one dimensional curve trees, each of which has their own Hausdorff dimension, jointly creating a smooth dimension field. We extend the two dimensional surface model to arbitrary dimensions $n$ where $n-1$ interface manifolds transport the $n$ field state of the parent patches to their child branches. We note that the balance or discrepancy between patch field dimension and the dimensions in which the branches may evolve, determine the analytical regime from essentially geometrical to essentially operational.