The dynamics of Super-Apollonian continued fractions
| Autor: | Elena Fuchs, Robert Hines, Katherine E. Stange, Sneha Chaubey |
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| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Dynamical systems theory Cayley graph Applied Mathematics General Mathematics 010102 general mathematics 01 natural sciences Pythagorean triple Ergodic theory Invariant measure 0101 mathematics Generalized continued fraction Dynamical system (definition) Gaussian rational Mathematics |
| Zdroj: | Transactions of the American Mathematical Society. 372:2287-2334 |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/tran/7372 |
| Popis: | We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks, and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are “reflective” versions of the complex continued fractions of A. L. Schmidt. They also describe a reduction algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean triples. For these dynamical systems, we produce an invertible extension and an invariant measure, which we conjecture is ergodic. We consider some statistics of the related continued fraction expansions, and we also examine the restriction of these systems to the real line, which gives a reflective version of the usual continued fraction algorithm. Finally, we briefly consider an alternate setup corresponding to a tree of Lorentz quadruples ordered by arithmetic complexity. |
| Databáze: | OpenAIRE |
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