Exact dimensional for Bernoulli measures and the Gauss map

A new proof of the dimension gap for the Gauss map

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000104 ◽

2021 ◽

pp. 1-29

Author(s):

NATALIA JURGA

Keyword(s):

Lower Bounds ◽

Asymptotic Variance ◽

Thermodynamic Formalism ◽

Gauss Map ◽

Probability Vector ◽

Bernoulli Measure ◽

Bernoulli Measures

Abstract In [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a ‘dimension gap’ meaning that for some c > 0, sup p dim μ p < 1– c, where μp denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

Download Full-text

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Forum of Mathematics Sigma ◽

10.1017/fms.2021.10 ◽

2021 ◽

Vol 9 ◽

Author(s):

Joseph Malkoun ◽

Peter J. Olver

Keyword(s):

Convex Hull ◽

Convex Geometry ◽

Gauss Map ◽

Parameter Family ◽

Convex Hulls ◽

Dimensional Sphere ◽

Continuous Maps ◽

Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

Download Full-text

A note on surfaces with prescribed oriented Euclidean Gauss map

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.537 ◽

2005 ◽

Vol 2005 (4) ◽

pp. 537-543

Author(s):

Ricardo Sa Earp ◽

Eric Toubiana

Keyword(s):

Euclidean Space ◽

Hyperbolic Space ◽

Gauss Map ◽

Compatibility Relation ◽

Hyperbolic Gauss Map ◽

Conformal Immersion

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

Download Full-text