A dynamic version of Schur’s problem for rotations on the unit interval

Abstract

We introduce a dynamical analog of Schur’s problem by defining the dynamic Schur number S(k, f), the smallest integer n such that every k-coloring of a space Y and every orbit segment On(x0,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}_n(x_0,f)$$\end{document} under a transformation f:Y→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :Y\rightarrow Y$$\end{document} contains a monochromatic triple {fa(x0),fb(x0),fa+b(x0)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{f^a(x_0),f^b(x_0),f^{a+b}(x_0)\}$$\end{document} with a,b,a+b≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b,a+b \le n$$\end{document}. This framework extends classical additive Ramsey theory to dynamical systems. We establish general bounds relating S(k, f) to the classical Schur numbers and determine some exact values for circle rotations Tα(x)=x+α(mod1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\alpha (x)=x+\alpha \pmod 1$$\end{document} under uniform and refined partitions. Computational experiments based on a late acceptance Hill-Climbing heuristic recover all known Schur numbers in the dynamic setting, motivating further exploration through combinatorial dynamics.