Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part, number of distinct parts and number of parts of multiplicity k, all accurate to o(1).Comment: 28 page

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory-Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.Comment: 67 pages, 21 figures; v2: changes to sections 10.2-10.3, improved figures; v3: additional figures and minor updates, add reference to IPCO abstract. CC-BY-S

Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis

For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution)

Dynamics of Modular Matings

In the paper 'Mating quadratic maps with the modular group II' the current authors proved that each member of the family of holomorphic $(2:2)$ correspondences $\mathcal{F}_a$: $$\left(\frac{az+1}{z+1}\right)^2+\left(\frac{az+1}{z+1}\right)\left(\frac{aw-1}{w-1}\right) +\left(\frac{aw-1}{w-1}\right)^2=3,$$ introduced by the first author and C. Penrose in 'Mating quadratic maps with the modular group', is a mating between the modular group and a member of the parabolic family of quadratic rational maps $P_A:z\to z+1/z+A$ whenever the limit set of $\mathcal{F}_a$ is connected. Here we provide a dynamical description for the correspondences $\mathcal{F}_a$ which parallels the Douady and Hubbard description for quadratic polynomials. We define a B\"ottcher map and a Green's function for $\mathcal{F}_a$, and we show how in this setting periodic geodesics play the role played by external rays for quadratic polynomials. Finally, we prove a Yoccoz inequality which implies that for the parameter $a$ to be in the connectedness locus $M_{\Gamma}$ of the family $\mathcal{F}_a$, the value of the log-multiplier of an alpha fixed point which has combinatorial rotation number $1/q$ lies in a strip whose width goes to zero at rate proportional to $(\log q)/q^2$