Spectral and dynamical results related to certain non-integer base expansions on the unit interval

TL;DR

This work analyzes spectral and dynamical properties of β-expansions in Parry type through the transfer operator $\mathcal{P}$ and the Koopman operator $\mathfrak{K}$ induced by the base-$β$ map $T_β$ on the unit interval. It proves that Lipschitz observables funnel exponentially fast in $L^1$ to the invariant density $u_1$ associated with the eigenvalue $1$, while the eigenvalue $1$ is non-isolated and the entire open unit disk lies in the point spectrum of $\mathcal{P}$ for $1\le p\le 2$, with explicit eigenfunctions $\psi_z$ for $|z|<1$. A key insight is that the weighted Koopman operator $\widetilde{\mathfrak{K}}=u_1^{1/p}\,\mathfrak{K}\,u_1^{-1/p}$ is an isometry whose spectrum equals $\overline{\mathbb{D}}$, allowing the construction of $\psi_z$ via a Neumann-series and a finite-dimensional invariant subspace analysis. The results yield exponential decay of correlations and ergodicity for $T_β$, with implications for a strong law of large numbers for observables along the β-map, and they connect non-integer-base dynamics with precise spectral portraits of the associated transfer and Koopman operators.

Abstract

We consider certain non-integer base $β$-expansions of Parry's type and we study various properties of the transfer (Perron-Frobenius) operator $\mathcal{P}:L^p([0,1])\mapsto L^p([0,1])$ with $p\geq 1$ and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these $β$-expansions. We show that if $f$ is Lipschitz, then the iterated sequence $\{\mathcal{P}^N f\}_{N\geq 1}$ converges exponentially fast (in the $L^1$ norm) to an invariant state corresponding to the eigenvalue $1$ of $\mathcal{P}$. This "attracting" eigenvalue is not isolated: for $1\leq p\leq 2$ we show that the point spectrum of $\mathcal{P}$ also contains the whole open complex unit disk and we explicitly construct some corresponding eigenfunctions.

Figures (2)

• Figure 1: Illustration of the map $\psi_0$
• Figure 2: The first layer

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma A.1