Directional derivatives and the central limit theorem on compact general one-dimensional lattices

TL;DR

This paper proves a central limit theorem for Lipschitz-Gibbs probabilities on compact one-dimensional lattices, using properties of the Ruelle operator and analyzing derivatives of entropy and pressure.

Contribution

It establishes the CLT in a general lattice setting and provides explicit estimates of directional derivatives of entropy and pressure, including conditions for maximal derivatives.

Findings

Proves CLT for Lipschitz-Gibbs probabilities on compact lattices.

Provides explicit estimates for first and second directional derivatives of entropy.

Identifies conditions for maximal entropy derivative values in the kernel of the Ruelle operator.

Abstract

We will show the central limit theorem for the general one-dimensional lattice where the space of symbols is a compact metric space. We consider the CLT for Lipschitz-Gibbs probabilities and in the proof we use several properties of the Ruelle operator defined on our setting; this will require fixing an {\em a priori probability}. An important issue in the proof of the CLT is the existence of a certain second-order derivative, and this will follow from the analytic properties that will be described in detail throughout the paper. As additional results of independent interest, we will also describe some explicit estimates of the first and second directional derivatives of some dynamical entities like entropy and pressure. For example: given a fixed potential $f$, and a variable observable $\eta$ on the Kernel of the Ruelle operator $\mathcal{L}_f$, we consider the equilibrium probability $\mu_{f + t \,\eta}$ for $f + t \,\eta$. We estimate the values $ \frac{d}{dt} h (\mu_{f + t \,\eta})|_{t=0}$ and $ \frac{d^2}{dt^2} h (\mu_{f + t \,\eta})|_{t=0}$, where $h (\mu_{f + t \,\eta})$ is the entropy of $ \mu_{C + t \,\eta}$. For fixed $f$ we can find conditions that can indicate the $\eta$ attaining the maximal possible value of $ \frac{d}{dt} h (\mu_{f + t \,\eta})|_{t=0}$ (up to a natural normalization of $\eta)$, entirely in terms of elements on the kernel of $\mathcal{L}_f$. We also consider directional derivatives of the eigenfunction.