[Paper Review] Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

Artur O. Lopes, Ali Messaoudi+2 more|arXiv (Cornell University)|Oct 10, 2019

Advanced Banach Space Theory28 references4 citations

TL;DR

This paper establishes the existence of invariant, ergodic, and fully supported σ-additive probability measures for weighted shift operators on non-reflexive Banach spaces such as $c_0(\mathbb{R})$ and $l^p(\mathbb{R})$ ($1 \leq p < \infty$) using a novel adaptation of thermodynamic formalism. By introducing a transfer operator $\mathcal{L}_A$ for Hölder continuous potentials and leveraging a priori measures on the kernel, the authors prove the existence of a unique Gibbs state via the Ruelle-Perron-Frobenius theorem, extending results beyond reflexive spaces.

ABSTRACT

We show the existence of invariant ergodic $σ$-additive probability measures with full support on $X$ for a class of linear operators $L: X o X$, where $L$ is a weighted shift operator and $X$ either is the Banach space $c_0(\mathbb{R})$ or $l^p(\mathbb{R})$ for $1\leq p

Motivation & Objective

• To establish the existence of invariant, ergodic, and fully supported σ-additive probability measures for discrete-time linear dynamics on non-reflexive Banach spaces such as $c_0(\mathbb{R})$ and $l^p(\mathbb{R})$.
• To extend the applicability of thermodynamic formalism—previously limited to compact or reflexive settings—to a broad class of linear operators with non-trivial kernels.
• To provide a new, non-approximation-based method for constructing Gibbs measures in infinite-dimensional linear dynamics, differing from standard ergodic theory techniques.
• To characterize conditions under which a priori measures on the kernel of a linear operator are 'adapted', ensuring convergence of the dual transfer operator to a unique fixed point.

Proposed method

• Adapts the Ruelle-Perron-Frobenius theorem to a transfer operator $\mathcal{L}_A$ acting on continuous functions over $X = c_0(\mathbb{R})$ or $l^p(\mathbb{R})$, defined via a bounded Hölder continuous potential $A$.
• Introduces a dual operator $\mathcal{L}_A^*$ acting on the 1-Wasserstein space of probability measures on $X$, ensuring convergence to a unique fixed point.
• Requires an a priori probability measure $m$ on the kernel of the linear operator $L$, with conditions for $m$ to be 'adapted'—ensuring tightness and convergence of the iterated dual operator.
• Uses Prokhorov’s theorem and tightness estimates to prove sequential compactness of the orbit of the dual operator, leading to existence of a limit measure.
• Applies a contraction argument in the Wasserstein distance to show that the dual operator's iterates converge to a unique fixed point, which is the Gibbs measure.
• Establishes criteria for adapted measures using polynomial and exponential tail conditions on $m$, ensuring summability conditions required for convergence.

Experimental results

Research questions

• RQ1 Can invariant, ergodic, and fully supported probability measures be constructed for weighted shift operators on non-reflexive Banach spaces like $c_0(\mathbb{R})$ and $l^1(\mathbb{R})$ using thermodynamic formalism?
• RQ2 Under what conditions on the a priori measure $m$ defined on the kernel of $L$ does the dual transfer operator $\mathcal{L}_A^*$ converge to a unique fixed point?
• RQ3 How can the Ruelle-Perron-Frobenius theorem be adapted to infinite-dimensional, non-compact settings where standard compactness fails?
• RQ4 What are the sufficient conditions on the growth of the weights $d_n$ and the tail behavior of $m$ to ensure the existence of a Gibbs state?

Key findings

• A unique Gibbs probability measure exists as the fixed point of the dual transfer operator $\mathcal{L}_A^*$ on the 1-Wasserstein space of probabilities over $X = c_0(\mathbb{R})$ or $l^p(\mathbb{R})$, $1 \leq p < \infty$.
• The constructed Gibbs measure is invariant, ergodic, and has full support on $X$, even when $X$ is non-reflexive, such as $c_0(\mathbb{R})$ or $l^1(\mathbb{R})$.
• The existence of the Gibbs measure is guaranteed when the a priori measure $m$ on the kernel is adapted, which holds if $m$ has polynomial tails of order $\gamma > 1$ and $n^\ell / d_n \in X$ for some $\ell > \gamma^{-1}$.
• For exponential tails, $m$ is adapted if $((\log n)/d_n) \in X$, which is satisfied when $d_n$ grows exponentially.
• The convergence of the dual operator $\mathcal{L}_A^*$ to the Gibbs measure is proven via a contraction argument in the Wasserstein distance, showing that the distance between iterates of any two initial measures tends to zero.
• The method provides a new, non-approximation-based construction of invariant measures, differing from prior approaches that relied on dense orbits in reflexive spaces.