Arithmetic Dynamics

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the circle; (3) Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodic-theoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create "redundant" representations (those whose space of "digits" is a priori larger than necessary) and study their combinatorics.Comment: 45 pages in Latex + 3 figures in ep

Semiregular automorphisms of vertex-transitive graphs of certain valencies

AbstractIt is shown that a vertex-transitive graph of valency p+1, p a prime, admitting a transitive action of a {2,p}-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605–615])

Numerical calculation of three-point branched covers of the projective line

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangulations include the Brieskorn homology spheres $\Sigma (p,q,r)$, the lens spaces $\operatorname{L} (q,1)$ and, as a limit case, $(\mathbf{S}^2 \times \mathbf{S}^1)^{\# (p-1)(q-1)}$.Comment: 28 pages, 9 figures. Minor update. To appear in Israel Journal of Mathematic

Random subgraphs of finite graphs: I. The scaling window under the triangle condition

We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their degree. We define the critical threshold $p_c=p_c(\mathbb G,\lambda)$ to be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $\lambda V^{1/3}$, where $\lambda$ is fixed and positive. We show that for any such model, there is a phase transition at $p_c$ analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold $p_c$. In particular, we show that the largest cluster inside a scaling window of size |p-p_c|=\Theta(\cn^{-1}V^{-1/3}) is of size $\Theta(V^{2/3})$, while below this scaling window, it is much smaller, of order $O(\epsilon^{-2}\log(V\epsilon^3))$, with \epsilon=\cn(p_c-p). We also obtain an upper bound O(\cn(p-p_c)V) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order \Theta(\cn(p-p_c)). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the $n$-cube and certain Hamming cubes, as well as the spread-out $n$-dimensional torus for $n>6$

Spectral Fundamentals and Characterizations of Signed Directed Graphs

The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts. To represent such signed directed graphs, we use a striking equivalence to $\mathbb{T}_6$-gain graphs to formulate a Hermitian adjacency matrix, whose entries are the unit Eisenstein integers $\exp(k\pi i/3),$ $k\in \mathbb{Z}_6.$ Many well-known results, such as (gain) switching and eigenvalue interlacing, naturally carry over to this paradigm. We show that non-empty signed directed graphs whose spectra occur uniquely, up to isomorphism, do not exist, but we provide several infinite families whose spectra occur uniquely up to switching equivalence. Intermediate results include a classification of all signed digraphs with rank $2,3$, and a deep discussion of signed digraphs with extremely few (1 or 2) non-negative (eq. non-positive) eigenvalues