Ultrametric Lindenmayer systems over prime alphabets: p-adic fractals and boundary dynamics

Parallel rewriting in the sense of Lindenmayer generates branching patterns whose prefix trees carry a natural ultrametric. For a prime p, we introduce ultrametric Lindenmayer systems of prime type p (ULₚS) by identifying the p-ary prefix tree with the tree of basic balls in Zₚ and by taking labeled local refinement rules as the primary data. Every ULₚS canonically determines a graph-directed iterated function system on Zₚ, and its limit set Lambda (S) is a p-adic path set fractal. The disjointness of the cosets d + pZₚ makes the open set condition automatic; under irreducibility and non-emptiness, dimH Lambda (S) = log rho (Mₐcc) / log p. The transition graph also yields grammar-level criteria for non-emptiness, surjectivity, perfection, and reducibility. We further study constant-length substitutions sigma: Sigmaₚ -> Sigmaₚᵏ through the induced boundary maps Fₛigma: Zₚ -> Zₚ. These maps satisfy a k-power Hölder law, with equality for all distinct points precisely in the prefix-injective case, and their images have Hausdorff dimension log m / (k log p), where m is the number of distinct substituted blocks. Examples include Fibonacci-type grammars, Thue-Morse, a 3-adic Cantor set, and a schematic coral rule.