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Period-Doubling Cascades Invariants: Braided Routes To Chaos

Eran Igra, Valerii Sopin

TL;DR

This work develops a braid-theoretic framework for describing how dynamical systems transition from orderly behavior to chaos via period-doubling cascades under isotopy. It introduces three invariants—the Conformal Index, the Index-Invariant, and the Trace-Invariant—that translate cascades into extremal quasiconformal data and representation-theoretic quantities, enabling both qualitative and computable analyses. The authors apply these tools to concrete systems such as Henon maps and Shilnikov-type perturbations, deriving symbolic dynamics and lower bounds on complexity, while also outlining numerical schemes to approximate the invariants and discussing connections to number theory and Teichmüller theory. The results suggest that braid-theoretic invariants capture essential topological structure of forcing dynamics and offer a path to unifying dynamical systems with areas like knot theory, higher Teichmüller theory, and quantum invariants.

Abstract

By a classical result of Kathleen Alligood and James Yorke we know that as we isotopically deform a map $f:ABCD\to\mathbb{R}^2$ to a Smale horseshoe map we should often expect the dynamical complexity to increase via a period--doubling route to chaos. Inspired by this fact and by how braids force the existence of complex dynamics, in this paper we introduce three topological invariants that describe the topology of period--doubling routes to chaos. As an application, we use our methods to ascribe symbolic dynamics to perturbations of the Shilnikov homoclinic scenario and to study the dynamics of the Henon map.

Period-Doubling Cascades Invariants: Braided Routes To Chaos

TL;DR

This work develops a braid-theoretic framework for describing how dynamical systems transition from orderly behavior to chaos via period-doubling cascades under isotopy. It introduces three invariants—the Conformal Index, the Index-Invariant, and the Trace-Invariant—that translate cascades into extremal quasiconformal data and representation-theoretic quantities, enabling both qualitative and computable analyses. The authors apply these tools to concrete systems such as Henon maps and Shilnikov-type perturbations, deriving symbolic dynamics and lower bounds on complexity, while also outlining numerical schemes to approximate the invariants and discussing connections to number theory and Teichmüller theory. The results suggest that braid-theoretic invariants capture essential topological structure of forcing dynamics and offer a path to unifying dynamical systems with areas like knot theory, higher Teichmüller theory, and quantum invariants.

Abstract

By a classical result of Kathleen Alligood and James Yorke we know that as we isotopically deform a map to a Smale horseshoe map we should often expect the dynamical complexity to increase via a period--doubling route to chaos. Inspired by this fact and by how braids force the existence of complex dynamics, in this paper we introduce three topological invariants that describe the topology of period--doubling routes to chaos. As an application, we use our methods to ascribe symbolic dynamics to perturbations of the Shilnikov homoclinic scenario and to study the dynamics of the Henon map.

Paper Structure

This paper contains 24 sections, 21 theorems, 8 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Braid groups
  4. Braids
  5. Left-invariant linear ordering
  6. Linear representations
  7. Index
  8. The Burau representation
  9. The Special linear group
  10. Index of subgroups of the special linear group
  11. Links to Number Theory
  12. Continued fractions
  13. P-adics
  14. Forcing theory and the Betsvina-Handel algorithm
  15. Braiding routes to chaos
  16. ...and 9 more sections

Key Result

Theorem 1

Given any $N\geq2$, let denote $\mathbb{Z}_N$ the ring of integers modulo $N \geq 2$. Then, the natural map $SL(n, \mathbb{Z}) \longrightarrow SL(n, \mathbb{Z}_N)$ defined by replacing each entry $a_{i,j}$ of $A\in SL(n,\mathbb{Z})$ with ${a_{i,j}} \mod N$ is onto.

Figures (13)

  • Figure 1: A braid corresponding to a periodic orbit of minimal period 8 (a braid on 8 strands) for the $\cap$-Smale horseshoe (see Figure 42 in $[54]$).
  • Figure 2: Braids in $B_2$ - on the upper left side we have the identity braid, on the upper right side we have a braid and its inverse, and on the bottom we have an example of a non-trivial braid generated by concatenating two braids.
  • Figure 3: The cabling operation.
  • Figure 4: On the right - a disc diffeomorphism that permutes the red and green points and fixes the blue points, on the left - the action on the purple and yellow arcs (i.e., the dashed lines) forces the creation of a braid. We denote this braid by Russian G:$\hbox{\CYRZH}$.
  • Figure 5: The homeomorphisms $h$ (on the left) and $g$ (on the right) are isotopic on $D$$rel$$\gamma$, where $\gamma=\{x_1,x_2,x_3\}$ (in this scenario, $h(x_1)=g(x_1)=x_2$, $h(x_2)=g(x_2)=x_1$ and $x_3$ remains fixed). This is exemplified by how they distort the green curves connecting the elements in $\gamma$ (i.e., the spine of $S$) - whose respective images under $h$ and $g$ are the dashed red lines.
  • ...and 8 more figures

Theorems & Definitions (40)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 4
  • Definition 5
  • ...and 30 more