Cartographic veracity in medieval mapping: analyzing geographical variation in the Gough Map of Great Britain

This article explores statistical approaches for assessing the relative accuracy of medieval mapping. It focuses on one particular map, the Gough Map of Great Britain. This is an early and remarkable example of a medieval “national” map covering Plantagenet Britain. Conventionally dated to c. 1360, the map shows the position of places in and coastal outline of Great Britain to a considerable degree of spatial accuracy. In this article, aspects of the map's content are subjected to a systematic analysis to identify geographical variations in the map's veracity, or truthfulness. It thus contributes to debates among historical geographers and cartographic historians on the nature of medieval maps and mapping and, in particular, questions of their distortion of geographic space. Based on a newly developed digital version of the Gough Map, several regression-based approaches are used here to explore the degree and nature of spatial distortion in the Gough Map. This demonstrates that not only are there marked variations in the positional accuracy of places shown on the map between regions (i.e., England, Scotland, and Wales), but there are also fine-scale geographical variations in the spatial accuracy of the map within these regions. The article concludes by suggesting that the map was constructed using a range of sources, and that the Gough Map is a composite of multiscale representations of places in Great Britain. The article details a set of approaches that could be transferred to other contexts and add value to historic maps by enhancing understanding of their contents.

Analyticity of smooth eigenfunctions and spectral analysis of the Gauss map

We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x) = integral a(x, y) f(b( x, y)) mu(dy) acting on functions f: [u, v] --> C ( evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius-Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius-Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set C\(-infinity, 1]. Analyzing the spectrum of the Frobenius Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Ropstorff asymptotic formula for the decay of correlations of the Gauss map.

The resonance spectrum of the cusp map in the space of analytic functions

We prove that the Frobenius-Perron operator $U$ of the cusp map $F:[-1,1]\to [-1,1]$, $F(x)=1-2 x^{1/2}$ (which is an approximation of the Poincare section of the Lorenz attractor) has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1. We also prove that for any $q\in (0,1)$ the spectrum of $U$ in the Hardy space in the disk $\{z\in C:|z-q|

Extended spectral decompositions of the Renyi map

We present new general methods to obtain spectral decompositions of dynamical systems in rigged Hilbert spaces and investigate the existence of resonances and the completeness of the associated eigenfunctions. The results are illustrated explicitly for the simplest chaotic endomorphism, namely the Renyi map.