Dynamical properties of the Pascal adic transformation

We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a two-symbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing

All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs

For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a minimum $st$-cut in $G$? We solve this problem in preprocessing time $O(n\log^3 n)$ for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space $O(n)$ that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is $2^{O(g^2)}$