Decay of distance autocorrelation and Lyapunov exponents

This work presents numerical evidences that for discrete dynamical systems with one positive Lyapunov exponent the decay of the distance autocorrelation is always related to the Lyapunov exponent. Distinct decay laws for the distance autocorrelation are observed for different systems, namely exponential decays for the quadratic map, logarithmic for the H\'enon map and power-law for the conservative standard map. In all these cases the decay exponent is close to the positive Lyapunov exponent. For hyperbolic conservative systems, the power-law decay of the distance autocorrelation tends to be guided by the smallest Lyapunov exponent.Comment: 7 pages, 8 figure

The Euclidean distance degree of an algebraic variety

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic

Mapping the Milky Way bulge at high resolution: the 3D dust extinction, CO, and X factor maps

Three dimensional interstellar extinction maps provide a powerful tool for stellar population analysis. We use data from the VISTA Variables in the Via Lactea survey together with the Besan\c{c}on stellar population synthesis model of the Galaxy to determine interstellar extinction as a function of distance in the Galactic bulge covering $-10 < l < 10$ and $-10 < b <5$. We adopted a recently developed method to calculate the colour excess. First we constructed the H-Ks vs. Ks and J-Ks vs. Ks colour-magnitude diagrams based on the VVV catalogues that matched 2MASS. Then, based on the temperature-colour relation for M giants and the distance-colour relations, we derived the extinction as a function of distance. The observed colours were shifted to match the intrinsic colours in the Besan\c{c}on model as a function of distance iteratively. This created an extinction map with three dimensions: two spatial and one distance dimension along each line of sight towards the bulge. We present a 3D extinction map that covers the whole VVV area with a resolution of 6' x 6', using distance bins of 0.5 kpc. The high resolution and depth of the photometry allows us to derive extinction maps for a range of distances up to 10 kpc and up to 30 magnitudes of extinction in $A_{V}$. Integrated maps show the same dust features and consistent values as other 2D maps. We discuss the spatial distribution of dust features in the line of sight, which suggests that there is much material in front of the Galactic bar, specifically between 5-7 kpc. We compare our dust extinction map with high-resolution $\rm ^{12}CO$ maps towards the Galactic bulge, where we find a good correlation between $\rm ^{12}CO$ and $\rm A_{V}$. We determine the X factor by combining the CO map and our dust extinction map. Our derived average value is consistent with the canonical value of the Milky Way.Comment: 11 pages, 18 figures, accepted for publication in Astronomy&Astrophysic

Sets of invariant measures and Cesaro stability

Sets of invariant measures are considered for continuous maps of a metric compact set. We take Kantorovich metric to calculate distance between measures and Hausdorff metrics to calculate distance between compact sets. Consider the function that makes correspondence between a continuous map and the set of all its Borel probability invariant measures. We demonstrate that a typical map is a continuity point of that function. Using approaches of Takens' tolerance stability theory we provide some corollaries that demonstrate that for a typical map points are structurally stable in a statistical sense.Comment: 11 pages, no figure