scholarly journals The Schottky–Klein prime function and counting functions for Fenchel double crosses

2021 ◽  
Author(s):  
M. Pollicott
Keyword(s):  
Nineteenth Century ◽  
Ergodic Theory ◽  
Hyperbolic Geometry ◽  
Complex Analysis ◽  
Thermodynamic Formalism ◽  
Counting Problem ◽  
Counting Functions ◽  
Prime Function

AbstractWe relate the classical nineteenth century Schottky–Klein function in complex analysis to a counting problem for pairs of geodesics in hyperbolic geometry studied by Fenchel. We then solve the counting problem using ideas from ergodic theory and thermodynamic formalism.

Lattice orbit distribution on ℝ2

2009 ◽  
Vol 30 (4) ◽  
pp. 1201-1214 ◽  
Author(s):  
ARNALDO NOGUEIRA
Keyword(s):  
Ergodic Theory ◽  
Diophantine Approximation ◽  
Error Term ◽  
Absolute Error ◽  
Compact Set ◽  
Linear Action ◽  
Counting Problem ◽  
Asymptotical Behaviour ◽  
Diophantine Property ◽  
The Absolute

AbstractWe study the distribution on ℝ2 of the orbit of a vector under the linear action of SL(2,ℤ). Let Ω⊂ℝ2 be a compact set and x∈ℝ2. Let N(k,x) be the number of matrices γ∈SL(2,ℤ) such that γ(x)∈Ω and ‖γ‖≤k, k=1,2,…. If Ω is a square, we prove the existence of an absolute error term for N(k,x), as k→∞, for almost every x, which depends on the Diophantine property of the ratio of the coordinates of x. Our approach translates the question into a Diophantine approximation counting problem which provides the absolute error term. The asymptotical behaviour of N(k,x) is also obtained using ergodic theory.

scholarly journals AnL 1 counting problem in ergodic theory

2005 ◽  
Vol 95 (1) ◽  
pp. 221-241 ◽  
Author(s):  
Idris Assani ◽  
Zoltán Buczolich ◽  
R. Daniel Mauldin
Keyword(s):  
Ergodic Theory ◽  
Counting Problem

scholarly journals On the Non-Holonomic Character of Logarithms, Powers, and the $n$th Prime Function

10.37236/1894 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Philippe Flajolet ◽  
Stefan Gerhold ◽  
Bruno Salvy
Keyword(s):  
Differential Equations ◽  
Complex Analysis ◽  
Structure Theorem ◽  
Prime Numbers ◽  
Fractional Powers ◽  
Open Problems ◽  
Naturally Occurring ◽  
Abelian Theorem ◽  
Linear Differential ◽  
Prime Function

We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularity-based methods and several naturally occurring sequences are proved to be non-holonomic.

The Perils of Parallels!

2020 ◽  
pp. 156-166
Author(s):  
Nicholas Mee
Keyword(s):  
Nineteenth Century ◽  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Spherical Geometry ◽  
Space And Time ◽  
The Earth

In the nineteenth century, three mathematicians—Bolyai, Gauss, and Lobachevsky—almost simultaneously discovered the possibility of non-Euclidean or hyperbolic geometries. These geometries rest on axioms that do not include the parallel postulate. This means that many results of Euclidean geometry do not hold. Spherical geometry is considered as a model to illustrate why this is the case. The mathematician Donald Coxeter inspired artist M. C. Escher to produce remarkable artworks based on the hyperbolic geometry of the Poincaré disc. Gauss attempted to measure the curvature of the space around the Earth. Since Einstein, we know that gravity curves space and time.

scholarly journals Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1268
Author(s):  
Azmeer Nordin ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Zeta Function ◽  
Complex Analysis ◽  
Discrete Dynamical System ◽  
Zeros Of Polynomials ◽  
Asymptotic Results ◽  
Square Roots ◽  
Shift Spaces ◽  
Closed Orbits ◽  
Counting Functions ◽  
Asymptotic Behaviours

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

A counting problem in ergodic theory and extrapolation for one-sided weights

2018 ◽  
Vol 134 (1) ◽  
pp. 237-254
Author(s):  
María J. Carro ◽  
María Lorente ◽  
Francisco J. Martín-Reyes
Keyword(s):  
Ergodic Theory ◽  
Counting Problem
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