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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

A Borel system $(X,S)$ is `almost Borel universal' if any free Borel dynamical system $(Y,T)$ of strictly lower entropy is isomorphic to a Borel subsystem of $(X,S)$, after removing a null set. We obtain and exploit a new sufficient condition for a topological dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a `generic' homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non-uniform specification implies almost Borel universality, and that $3$-colorings in $\mathbb Z^d$ and dimers in $\mathbb Z^2$ are almost Borel universal, Revised after Referee's comments; 70 pages

Graph homomorphisms from the $\mathbb{Z}^d$ lattice to $\mathbb{Z}$ are functions on $\mathbb{Z}^d$ whose gradients equal one in absolute value. These functions are the height functions corresponding to proper $3$-colorings of $\mathbb{Z}^d$ and, in two dimensions, corresponding to the $6$-vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures. Additional results are obtained in higher dimensions and include the fact that every Gibbs measure which is ergodic under even translations is extremal and that these Gibbs measures are stochastically ordered., Comment: 21 pages, 2 Figures. Corrected Lemma 4.2 (Lemma 4.12 in the current version) from the previous version