2区间分段仿射映射动力学与Hecke-Mahler级数

IF 0.7 1区数学 Q2 MATHEMATICS

Journal of Modern Dynamics Pub Date : 2019-07-19 DOI:10.3934/JMD.2021002

M. Laurent, A. Nogueira

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引用次数: 8

摘要

设$f : [0,1)\rightarrow [0,1)$为一个$2$ -区间分段仿射递增映射，它是内射但不是满射。这样的映射$f$有一个旋转数，可以用三个实数参数化。我们充分明确了$f$的动态，这要归功于两个特定的函数$\delta$和$\phi$，这取决于这些参数的定义涉及Hecke-Mahler系列。作为应用，我们证明了当三个参数为代数数时，$f$的旋转数是有理数。

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Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series

Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $\delta$ and $\phi$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.

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来源期刊

Journal of Modern Dynamics 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Modern Dynamics (JMD) is dedicated to publishing research articles in active and promising areas in the theory of dynamical systems with particular emphasis on the mutual interaction between dynamics and other major areas of mathematical research, including: Number theory Symplectic geometry Differential geometry Rigidity Quantum chaos Teichmüller theory Geometric group theory Harmonic analysis on manifolds. The journal is published by the American Institute of Mathematical Sciences (AIMS) with the support of the Anatole Katok Center for Dynamical Systems and Geometry at the Pennsylvania State University.