A Laplacian on the Full Shift Space

TL;DR

This paper develops a Laplacian operator on the one-sided shift space using difference operators, and introduces Green's functions to solve boundary value problems in this symbolic dynamical system.

Contribution

It extends rough analysis to the shift space by defining a Laplacian and associated Green's functions, providing new tools for analysis on symbolic dynamical systems.

Findings

Defined difference operators on the shift space

Constructed the Laplacian and Green's functions

Solved boundary value problems in the shift space

Abstract

In this paper, we consider the one-sided shift space on finitely many symbols and extend the theory of what is known as rough analysis. We define difference operators on an increasing sequence of subsets of the shift space that would eventually render the Laplacian on the space of real-valued continuous functions on the shift space. We then define the Green's function and the Green's operator that come in handy to solve the analogue to the Dirichlet boundary value problem on the shift space.