An approach to renormalization on the n-torus.

Area of Science:

• Number Theory
• Dynamical Systems
• Ergodic Theory

Background:

• The study revisits the coding theory of rotations and the continued fractions algorithm.
• It considers integer two-coloring with a specific proportion of colors.

Purpose of the Study:

• To define even n-coloring of integers.
• To develop a geometric approach to renormalization on tori.

Main Methods:

• Revisiting coding theory of rotations and their relation to flows.
• Analyzing the continued fractions algorithm via integer coloring.
• Defining and applying integer n-coloring.
• Utilizing linear flows on the n-torus.
• Employing first return maps on coding regions for renormalization.

Main Results:

• A novel definition of integer n-coloring is established.
• Rotations on the (n-1)-torus can be coded using linear flows on the n-torus.
• A simple geometric approach to renormalization on tori is achieved.

Conclusions:

• Integer n-coloring provides a unified framework for coding rotations and renormalization.
• The geometric approach offers new insights into the dynamics of tori and continued fractions.