Dirichlet eigenvalues of the Laplacian on full one-sided shift space

TL;DR

This paper investigates the spectral properties of the Laplacian on the full one-sided shift space by approximating it with finite subsets and analyzing the convergence of eigenvalues using spectral decimation.

Contribution

It provides a complete determination of the spectrum of difference operators approximating the Laplacian and establishes convergence of eigenvalues to the Laplacian's spectrum.

Findings

Spectrum of difference operators explicitly determined

Eigenvalues of difference operators converge to Laplacian eigenvalues

Spectral decimation method effectively applied to shift space

Abstract

The full one sided shift space over finite symbols is approximated by an increasing sequence of finite subsets of the space. The Laplacian on the space is then defined as a renormalised limit of the difference operators defined on these subsets. In this work, we determine the spectrum of these difference operators completely, using the method of spectral decimation. Further, we prove that under certain conditions, the renormalised eigenvalues of the difference operators converge to an eigenvalue of the Laplacian.