A Ternary (4k+1(2))/3 Collatz-Type Map and Its 4^r-Scaled Family
Year: 2026
Type: preprint
Abstract: This work presents and analyzes a ternary (4k+1(2))/3 Collatz-type map together with its 4^r-scaled family. The study develops a base ternary dynamical system and constructs a family of scaled maps indexed by integers r >= 0. The central structural result is an exact one-step scaling relation of the form Tr(4^r y) = 4^r T0(y), which establishes a direct correspondence between trajectories of the base system and those of its 4^r-scaled variants. As a consequence of this scaling relation, the corresponding dynamical structure is preserved across the family. In particular, periodic-orbit structure, stopping-time behavior, and basin membership are transported between the base map and its scaled copies. The paper also studies the associated 3-adic behavior and develops a heuristic geometric drift analysis. The resulting heuristic contraction factor is approximately 0.7698, corresponding to an average logarithmic drift consistent with contraction. This heuristic provides motivation for the convergence analysis but does not by itself constitute a proof of global convergence. The computational material accompanying the paper is provided for independent verification and reproducibility. It includes the source code used to test the conjugacy relation, definitions of the scaling rules, computational summaries for the base map, the LaTeX source, and the compiled article. The exact algebraic scaling relation is the principal mechanism connecting the members of the 4^r-scaled family: rather than defining independent dynamical systems with unrelated behavior, the scaled maps reproduce the corresponding base trajectories under multiplication by 4^r. Any global convergence statement discussed for the family should therefore be understood in relation to the corresponding convergence question for the base map. Exact scaling equivalence transfers the dynamical question between the systems, but it does not by itself establish global convergence.
Author: Farhad, Banazadeh
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211890823
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