Finite Resolution Dynamics

We develop a new mathematical model for describing a dynamical system at limited resolution (or finite scale), and we give precise meaning to the notion of a dynamical system having some property at all resolutions coarser than a given number. Open covers are used to approximate the topology of the phase space in a finite way, and the dynamical system is represented by means of a combinatorial multivalued map. We formulate notions of transitivity and mixing in the finite resolution setting in a computable and consistent way. Moreover, we formulate equivalent conditions for these properties in terms of graphs, and provide effective algorithms for their verification. As an application we show that the Henon attractor is mixing at all resolutions coarser than 10^-5.Comment: 25 pages. Final version. To appear in Foundations of Computational Mathematic

Countable locally 2-arc-transitive bipartite graphs

We present an order-theoretic approach to the study of countably infinite locally 2-arc-transitive bipartite graphs. Our approach is motivated by techniques developed by Warren and others during the study of cycle-free partial orders. We give several new families of previously unknown countably infinite locally-2-arc-transitive graphs, each family containing continuum many members. These examples are obtained by gluing together copies of incidence graphs of semilinear spaces, satisfying a certain symmetry property, in a tree-like way. In one case we show how the classification problem for that family relates to the problem of determining a certain family of highly arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page

Symmetry properties of subdivision graphs

The subdivision graph $S(\Sigma)$ of a graph $\Sigma$ is obtained from $\Sigma$ by `adding a vertex' in the middle of every edge of \Si. Various symmetry properties of $\S(\Sigma)$ are studied. We prove that, for a connected graph $\Sigma$, $S(\Sigma)$ is locally $s$-arc transitive if and only if $\Sigma$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. The diameter of $S(\Sigma)$ is $2d+\delta$, where $\Sigma$ has diameter $d$ and $0\leqslant \delta\leqslant 2$, and local $s$-distance transitivity of $\S(\Sigma)$ is defined for $1\leqslant s\leqslant 2d+\delta$. In the general case where $s\leqslant 2d-1$ we prove that $S(\Sigma)$ is locally $s$-distance transitive if and only if $\Sigma$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. For the remaining values of $s$, namely $2d\leqslant s\leqslant 2d+\delta$, we classify the graphs $\Sigma$ for which $S(\Sigma)$ is locally $s$-distance transitive in the cases, $s\leqslant 5$ and $s\geqslant 15+\delta$. The cases $\max\{2d, 6\}\leqslant s\leqslant \min\{2d+\delta, 14+\delta\}$ remain open