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On multidimensional infinite dihedral group extensions of Gibbs Markov maps

Jaime Gomez, Dalia Terhesiu

TL;DR

This work establishes a local central limit theorem for cocycles arising from multidimensional infinite dihedral group extensions $G_d$ of Gibbs Markov maps, addressing non-abelian and non-compact group actions. The authors develop a representation-theoretic framework via irreducible representations of $G_d$ and study twisted transfer operators $L_{\rho_\theta}$ to derive a Gaussian limit with covariance $\Sigma$, whose invertibility hinges on moment and moment-algebraic conditions. A phase transition in dimension $d$ is shown: for $d\le 2$ the group extension is mixing (ergodic) and yields Krickeberg mixing, while for $d\ge 3$ the extension is dissipative, with precise first-return time asymptotics that depend on $d$. The analysis combines a detailed perturbation theory around $\theta=0$ with a global inversion formula and a discrete renewal argument, producing a comprehensive LCLT for the cocycle $\psi_n$ and extending the LCLT for random walks on $G_d$ to the dynamical-systems setting. The results provide a rigorous quantitative description of how non-compact, non-abelian group extensions influence statistical properties of GM maps, including return-time statistics and mixing rates.

Abstract

We obtain a local central limit theorem for cocycles associated with a class of non abelian and non compact group extensions of Gibbs Markov maps. This class consists of multidimensional infinite dihedral groups. Unlike in the set up of the random walks on groups, we cannot use the convolution of measures on the group and instead we resort to an approach based on irreducible representations. Depending on the dimension of the group, we obtain either mixing, and thus ergodicity, or dissipativity. Also, we obtain the asymptotics of the first return time of the group extension to the origin.

On multidimensional infinite dihedral group extensions of Gibbs Markov maps

TL;DR

This work establishes a local central limit theorem for cocycles arising from multidimensional infinite dihedral group extensions of Gibbs Markov maps, addressing non-abelian and non-compact group actions. The authors develop a representation-theoretic framework via irreducible representations of and study twisted transfer operators to derive a Gaussian limit with covariance , whose invertibility hinges on moment and moment-algebraic conditions. A phase transition in dimension is shown: for the group extension is mixing (ergodic) and yields Krickeberg mixing, while for the extension is dissipative, with precise first-return time asymptotics that depend on . The analysis combines a detailed perturbation theory around with a global inversion formula and a discrete renewal argument, producing a comprehensive LCLT for the cocycle and extending the LCLT for random walks on to the dynamical-systems setting. The results provide a rigorous quantitative description of how non-compact, non-abelian group extensions influence statistical properties of GM maps, including return-time statistics and mixing rates.

Abstract

We obtain a local central limit theorem for cocycles associated with a class of non abelian and non compact group extensions of Gibbs Markov maps. This class consists of multidimensional infinite dihedral groups. Unlike in the set up of the random walks on groups, we cannot use the convolution of measures on the group and instead we resort to an approach based on irreducible representations. Depending on the dimension of the group, we obtain either mixing, and thus ergodicity, or dissipativity. Also, we obtain the asymptotics of the first return time of the group extension to the origin.
Paper Structure (22 sections, 12 theorems, 155 equations)

This paper contains 22 sections, 12 theorems, 155 equations.

Table of Contents

  1. Overview of the multidimensional infinite dihedral group
  2. The particular case $d=1$: the infinite dihedral group $D_\infty$.
  3. LCLT for the random walk (RW) on $G_d= \mathbb{Z}/2\mathbb{Z}\ltimes_\phi \mathbb{Z}^d$ (i.i.d. set up)
  4. Main ingredients of the proof
  5. Leading terms and final form of $\operatorname{Tr}( \rho_\theta(\pm 1,r) M(\theta)^n)$.
  6. Invertability of $\Sigma$.
  7. Completing the proof of Proposition \ref{['prop:rw']}
  8. Overview of Gibbs Markov maps and statement of LCLT
  9. Statement of the main results
  10. Inversion formula for LCLT in the Gibbs Markov set up
  11. Understanding $L_{\rho_\theta}$ acting on ${\mathcal{B}}_\beta$.
  12. Inversion formula
  13. Under \ref{['eq:assS']} the eigenvalues for $L_{0}^{\pm 1}$ are precisely $1/2$ and the corresponding eigenvectors are the same
  14. Understanding $L_{\rho_\theta}$ for $\theta$ close to $0_d$
  15. Completing the understanding of $L_{\rho_\theta}^n$ for $\theta\in B_\delta(0_d)=[-\delta,\delta]^d$
  16. ...and 7 more sections

Key Result

Proposition 1.1

The map $\Psi:[0,2\pi)^d\setminus T_d\to \widehat{G_d}$, given by $\Psi({\theta})=\rho_{{\theta}}$, is continuous, where the set $[0,2\pi)^d\setminus T_d$ is endowed with the usual topology inherited from $\mathbb{R}^d$ and $\widehat{G_d}$ with the Fell topology.

Theorems & Definitions (14)

  • Proposition 1.1
  • Lemma 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Lemma 6.1
  • Corollary 6.2
  • Lemma 6.3
  • ...and 4 more