A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a Pisot–Vijayaraghavan (PV) number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.

The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to obtain concrete answers in the edge-triangle model with fixed density of edges.

We report a study of resistive switching in a silicon-based memristor/resistive RAM (RRAM) device in which the active layer is silicon-rich silica. The resistive switching phenomenon is an intrinsic property of the silicon-rich oxide layer and does not depend on the diffusion of metallic ions to form conductive paths. Both unipolar and bipolar programming is demonstrated.

Switching exhibits the pinched hysteresis I/V loop characteristic of RRAM/memristive systems, and on/off resistance ratios of 104:1 or higher can be easily achieved. Scanning Tunnelling Microscopy suggests that switchable conductive pathways are 10nm in diameter or smaller.

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0–1 sequences (xk) such that xkx2k=0 for all k. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for multiplicative subshifts.

We investigate topological mixing for $\mathbb Z$ and $\mathbb R$ actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $\theta_2$ of the substitution matrix satisfies $|\theta_2|\ne 1$. If $|\theta_2|<1$, then (as is well known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|\theta_2|> 1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|\theta_2|=1$ is more delicate, and we only obtain some partial results.

For each irreducible hyperbolic automorphism $A$ of the $n$-torus we construct a sofic system $(\Sigma,\sigma)$ and a bounded-to-one continuous semiconjugacy from $(\Sigma,\sigma)$ to $({\Bbb T}^n,A)$. This construction is natural in the sense that it depends only on the characteristic polynomial of $A$ and, furthermore, it has an arithmetic interpretation.