Chaotic renormalization group flow and entropy gradients over Haros graphs

TL;DR

This paper explores the complex, chaotic renormalization group flow of Haros graphs, revealing stable fixed points, periodic orbits, and entropy behaviors, with implications for understanding RG dynamics in mathematical and physical systems.

Contribution

It introduces a detailed analysis of the RG flow over Haros graphs, identifying chaotic behavior, fixed points, and the entropy properties of periodic orbits, linking graph dynamics to number theory.

Findings

Chaotic RG flow with unstable periodic and aperiodic orbits.

A single stable fixed point with rational numbers as basin.

Constant entropy gradients within periodic RG orbits.

Abstract

Haros graphs have been recently introduced as a set of graphs bijectively related to real numbers in the unit interval. Here we consider the iterated dynamics of a graph operator $\cal R$ over the set of Haros graphs. This operator was previously defined in the realm of graph-theoretical characterisation of low-dimensional nonlinear dynamics, and has a renormalization group (RG) structure. We find that the dynamics of $\cal R$ over Haros graphs is complex and includes unstable periodic orbits or arbitrary period and non-mixing aperiodic orbits, overall portraiting a chaotic RG flow. We identify a single RG stable fixed point whose basin of attraction is the set of rational numbers, associate periodic RG orbits with (pure) quadratic irrationals and aperiodic RG orbits with (non-mixing) families of non-quadratic algebraic irrationals and trascendental numbers. Finally, we show that the entropy gradients inside periodic RG orbits are constant. We discuss the possible physical interpretation of such chaotic RG flow and speculate on the entropy-constant periodic orbits as a possible confirmation of a (quantum field-theoretic) $c$-theorem applied inside the invariant set of a RG flow.