The Octagonal PET I: Renormalization and Hyperbolic Symmetry

2012/09/11 by Richard Evan Schwartz, Schwartz, Richard Evan
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Protein Structure and Dynamics

paper · pdf · doi:10.48550/arxiv.1209.2390

openalex publication_date 2012/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a family of polytope exchange transformations (PETs) acting on parallelotopes in \R²ⁿ for n=1,2,3.... These PETs are constructed using a pair of lattices in \R²ⁿ. The moduli space of these PETs is GL_n(\R). We study the case n=1 in detail. In this case, we show that the 2-dimensional family is completely renormalizable and that the (2,4,∞) hyperbolic reflection triangle group acts (by linear fractional transformations) as the renormalization group on the moduli space. These results have a number of geometric corollaries for the system. Most of the paper is traditional mathematics, but some part of the paper relies on a rigorous computer-assisted proof involving integer calculations.

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