We provide a self-similar measure for the self-similar group G acting faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial z + i. We also provide an Lpresentation for G and calculations related to the spectrum of the Markov operator on the Schreier graph of the action of G on the orbit of a point on the boundary of the binary rooted tree.

We establish a generalization of the p-adic local monodromy theorem (of André, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called “fake annuli”. The latter correspond loosely to completions of a Laurent polynomial ring with respect to a valuation, which in this paper is restricted to be of monomial form; we defe...

A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group Bn . In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this paper is to understand this monodromy representation using standard tools from th...

We establish a generalization of the p-adic local monodromy theorem (of André, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called “fake annuli”. The latter correspond loosely to completions of a Laurent polynomial ring with respect to a monomial valuation. The result represents a step towards a higher-dimensiona...

For the fields depending on two of the four space-time coordinates only, the spaces of local solutions of various integrable reductions of Einstein’s field equations are shown to be the subspaces of the spaces of local solutions of the “null curvature” equations constricted by a requirement of a universal (i.e. solution independent) structures of the canonical Jordan forms of the unknown matrix...

In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with quasi-permutation monodromy representations outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of CP 1 . The solution is given in terms...

Let k be a field not of characteristic two and Λ be a set consisting of almost all rational primes invertible in k. Suppose we have a varietyX/k and strictly compatible system {M` → X : ` ∈ Λ} of constructible F`-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of M` is a subgroup of a corresponding isometry group Γ` over F`, and we say it h...

{ We demonstrate that an integrable approximation to the hydrogen atom in orthogonal electric and magnetic elds has monodromy, a fundamental dynamical property that makes a global deenition of action-angle variables and of quantum numbers impossible. When the eld strengths are suuciently small, we nd our integrable approximation using a two step normalization procedure. One of dynamically invar...