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Complexity of regular invertible p-adic motions
J. Pettigrew1, J. A. G. Roberts, F. Vivaldi
1Department of Mathematics, La Trobe University, VIC 3086, Australia.
Chaos (Woodbury, N.Y.)
|June 5, 2003
Summary
This study analyzes computational complexity in p-adic integer dynamics, focusing on Siegel discs. Researchers identified conditions determining cycle structure and proposed a minimal parametrization for maximal cycle length in quasi-periodic motions.
Area of Science:
- Number Theory
- Dynamical Systems
- Computational Complexity
Background:
- Quasi-periodic motions and Siegel discs are studied over p-adic integers.
- Invertible dynamics over integers modulo p(k) are generated by these systems.
- Key questions involve computing periods and orbit structure.
Purpose of the Study:
- To investigate computational complexity in p-adic dynamical systems.
- To identify conditions that determine the cycle structure of Siegel discs.
- To propose a minimal parametrization for specific tessellations.
Main Methods:
- Analysis of computational complexity in p-adic number systems.
- Study of polynomial maps and their dynamics.
- Application of number theory concepts, including Cebotarev's density theorem.
Main Results:
- Conditions were identified for cycle structure determination by the number of Siegel discs and associated parameters.
- A conjecture on minimal parametrization for a two-disc tessellation with maximal cycle length was proposed.
- The relevance of Cebotarev's density theorem to probabilistic descriptions was discussed.
Conclusions:
- The study provides insights into the computational aspects of quasi-periodic motions over p-adic integers.
- Understanding cycle structure is crucial for characterizing these dynamical systems.
- The findings contribute to the probabilistic analysis of p-adic dynamics.