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We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $q\le d+1$ , there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $q\ge d+2$ , any proper q-colouring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper q-colouring of the entire box. (3) When $q\geq 2d+1$ , the latter holds for any $n \ge 1$ . Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.

The classical Kirszbraun theorem says that all 1-Lipschitz functions f : A ⟶ R n , A ⊂ R n , with the Euclidean metric have a 1-Lipschitz extension to R n . For metric spaces X , Y we say that Y is X-Kirszbraun if all 1-Lipschitz functions f : A ⟶ Y , A ⊂ X , have a 1-Lipschitz extension to X. We analyze the case when X and Y are graphs with the usual path metric. We prove that Z d -Kirszbraun graphs are exactly graphs that satisfies a certain Helly's property . We also consider complexity aspects of these properties.

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk., Comment: Revised version after refereeing

Let $\mathcal H$ be a finite connected undirected graph and $\mathcal H_{walk}$ be the graph of bi-infinite walks on $\mathcal H$; two such walks $\{x_i\}_{i\in \mathbb Z}$ and $\{y_i\}_{i \in \mathbb Z}$ are said to be adjacent if $x_i$ is adjacent to $y_i$ for all $i \in \mathbb Z$. We consider the question: Given a graph $\mathcal H$ when is the diameter (with respect to the graph metric) of $\mathcal H_{walk}$ finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of $\mathbb Z^d$ with respect to the standard generators to $\mathcal H$) and are the subject of this paper., Comment: Corrected typo in the title on the Arxiv page

The Hammersley-Clifford theorem states that if the support of a Markov random field has a safe symbol then it is a Gibbs state with some nearest neighbour interaction. In this paper we generalise the theorem with an added condition that the underlying graph is bipartite. Taking inspiration from "Gibbs Measures and Dismantlable Graphs" by Brightwell and Winkler we introduce a notion of folding for configuration spaces called strong config-folding proving that if all Markov random fields supported on $X$ are Gibbs with some nearest neighbour interaction so are Markov random fields supported on the 'strong config-folds' and 'strong config-unfolds' of $X$., Comment: 27 pages, 7 figures; Typos corrected and some notation has been changed

Fix $d\geq 2$. Given a finite undirected graph ${\mathcal{H}}$ without self-loops and multiple edges, consider the corresponding ‘vertex’ shift, $\text{Hom}(\mathbb{Z}^{d},{\mathcal{H}})$, denoted by $X_{{\mathcal{H}}}$. In this paper, we focus on ${\mathcal{H}}$ which is ‘four-cycle free’. There are two main results of this paper. Firstly, that $X_{{\mathcal{H}}}$ has the pivot property, meaning that, for all distinct configurations $x,y\in X_{{\mathcal{H}}}$, which differ only at a finite number of sites, there is a sequence of configurations $x=x^{1},x^{2},\ldots ,x^{n}=y\in X_{{\mathcal{H}}}$ for which the successive configurations $x^{i},x^{i+1}$ differ exactly at a single site. Secondly, if ${\mathcal{H}}$ is connected ,then $X_{{\mathcal{H}}}$ is entropy minimal, meaning that every shift space strictly contained in $X_{{\mathcal{H}}}$ has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the ‘lifts’ of the configurations in $X_{{\mathcal{H}}}$ to the universal cover of ${\mathcal{H}}$ and the introduction of ‘height functions’ in this context.