Entropy Minimality of Certain Hom-Shifts

Abstract

In this thesis, we study structural and entropic properties of hom-shifts arising from finite connected graphs. Our main goal is to understand how the topology of the underlying graph influences rigidity phenomena in the associated shift space. We prove that whenever the square cover of a graph is a tree, the corresponding hom- shift is entropy minimal, meaning that forbidding any admissible pattern strictly decreases entropy. To develop the necessary tools, we investigate Lipschitz extension problems for graph homomorphisms and establish a bipartite version of the Kirszbraun–Helly equivalence, adapted to parity-preserving maps. This extension framework plays a key role in enabling global constructions from local constraints. A central component of our approach is the use of height functions obtained by lifting configurations using the square cover of the graph. This cover provides a natural setting in which lifts are well defined and distances encode global structural information. Using the ergodic theorems, we analyze the asymptotic growth of these height functions and show that maximal directional slope forces strong rigidity. A final chapter discusses a related direction for certain tiling systems, where we prove a necessary and sufficient condition for tileability.