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Twisted cocycle for interval exchange transformations: Invariant structures and Lyapunov spectrum

Hesam Rajabzadeh, Pedram Safaee

TL;DR

This work develops a comprehensive framework for the twisted cocycle over the toral extension of the Zorich renormalization for interval exchange transformations, revealing a detailed invariant-structure decomposition into covariant and invariant subbundles. It proves the Lyapunov spectrum is symmetric about zero and contains exactly $\kappa+1$ zero exponents, with $\kappa$ dictated by the Rauzy class, by constructing a family of invariant Hermitian and real symplectic forms. For rotation-type permutations, the twisted cocycle exhibits a degenerate spectrum, contrasting with the genus-$>1$ case where a positive exponent persists in the untwisted setting. The appendix extends these ideas to substitution dynamics, showing pure singular spectrum for a large class of two-letter substitutions and providing explicit examples that illustrate the invariant/covariant structures in practice. Together, these results link renormalization dynamics, spectral properties of IETs and translation flows, and substitution systems, with implications for local spectral dimensions and weak mixing phenomena.

Abstract

This paper investigates the algebraic and dynamical properties of the twisted cocycle, a $\mathrm{GL}(d, \mathbb{C})$-valued cocycle defined over the toral extension of the Zorich (Rauzy-Veech) renormalization for interval exchange transformations (IET). As a natural generalization of the Zorich cocycle, the twisted cocycle plays a central role in studying the asymptotic growth of twisted Birkhoff sums which in turn provide a suitable tool for obtaining fine spectral information about IETs and translation flows such as the local dimension of spectral measures and quantitative weak mixing. Although it shares similarities with the classical (untwisted) Zorich cocycle, structural differences make its analysis more challenging. Our results yield a block-form decomposition into invariant and covariant subbundles allowing us to demonstrate the existence of $κ+1$ zero exponents with respect to a large class of natural invariant measures where $κ$ is an explicit integer depending on the permutation. We establish the symmetry of the Lyapunov spectrum by showing the existence of a family of non-degenerate invariant symplectic forms. As a corollary, we prove that for rotation-type permutations, the twisted cocycle has a degenerate Lyapunov spectrum with respect to certain natural ergodic invariant measures in contrast with the higher genus case where the existence of at least one positive Lyapunov exponent is guaranteed by previous work of the authors. In the appendix, we apply our result about the invariant section to substitution systems and prove the pure singularity of the spectrum for a substantially large class of substitutions on two letters, while greatly simplifying the proof for some systems previously known to have this property.

Twisted cocycle for interval exchange transformations: Invariant structures and Lyapunov spectrum

TL;DR

This work develops a comprehensive framework for the twisted cocycle over the toral extension of the Zorich renormalization for interval exchange transformations, revealing a detailed invariant-structure decomposition into covariant and invariant subbundles. It proves the Lyapunov spectrum is symmetric about zero and contains exactly zero exponents, with dictated by the Rauzy class, by constructing a family of invariant Hermitian and real symplectic forms. For rotation-type permutations, the twisted cocycle exhibits a degenerate spectrum, contrasting with the genus- case where a positive exponent persists in the untwisted setting. The appendix extends these ideas to substitution dynamics, showing pure singular spectrum for a large class of two-letter substitutions and providing explicit examples that illustrate the invariant/covariant structures in practice. Together, these results link renormalization dynamics, spectral properties of IETs and translation flows, and substitution systems, with implications for local spectral dimensions and weak mixing phenomena.

Abstract

This paper investigates the algebraic and dynamical properties of the twisted cocycle, a -valued cocycle defined over the toral extension of the Zorich (Rauzy-Veech) renormalization for interval exchange transformations (IET). As a natural generalization of the Zorich cocycle, the twisted cocycle plays a central role in studying the asymptotic growth of twisted Birkhoff sums which in turn provide a suitable tool for obtaining fine spectral information about IETs and translation flows such as the local dimension of spectral measures and quantitative weak mixing. Although it shares similarities with the classical (untwisted) Zorich cocycle, structural differences make its analysis more challenging. Our results yield a block-form decomposition into invariant and covariant subbundles allowing us to demonstrate the existence of zero exponents with respect to a large class of natural invariant measures where is an explicit integer depending on the permutation. We establish the symmetry of the Lyapunov spectrum by showing the existence of a family of non-degenerate invariant symplectic forms. As a corollary, we prove that for rotation-type permutations, the twisted cocycle has a degenerate Lyapunov spectrum with respect to certain natural ergodic invariant measures in contrast with the higher genus case where the existence of at least one positive Lyapunov exponent is guaranteed by previous work of the authors. In the appendix, we apply our result about the invariant section to substitution systems and prove the pure singularity of the spectrum for a substantially large class of substitutions on two letters, while greatly simplifying the proof for some systems previously known to have this property.

Paper Structure

This paper contains 11 sections, 7 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. IETs and translation surfaces
  4. Rauzy-Veech and Zorich renormalizations
  5. Rauzy-Veech and Zorich cocycles
  6. Toral and twisted cocycles
  7. Invariant structures
  8. Invariant and covariant sections
  9. Invariant Hermitian and symplectic forms
  10. Proof of Main Theorem
  11. Appendix

Key Result

Corollary 1

Let $d>1$ be an integer and $\pi=(\pi_t, \pi_b)$ be a rotation type permutation. Then, the Lyapunov spectrum of the twisted cocycle with respect to the ergodic invariant measure $\mu \times m_{\mathbb{T}^2}$ is degenerate, i.e., all exponents are equal to zero.

Figures (2)

  • Figure 1: Renormalization of bottom type transforming the permutation $\left(ABCDEEDCBA\right)$ to the permutation $\left(AEBCDEDCBA\right).$
  • Figure 2: $S_0$ is the distinguished singularity and $c_{\pi, \eta}(C)$ is the amount of twist (induced by $-\eta$) along the colored edges.

Theorems & Definitions (19)

  • Corollary
  • Definition 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • proof
  • proof
  • ...and 9 more