scholarly journals Calculations of the Invariant Measure for Hurwitz Continued Fractions

2019 ◽  
pp. 1-13
Author(s):  
Ghaith Hiary ◽  
Joseph Vandehey
Keyword(s):  
Invariant Measure ◽  
Continued Fractions

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

SOME STRONG MIXING PROPERTIES OF A SEQUENCE OF RANDOM VARIABLES ARISING FROM α-CONTINUED FRACTIONS

2003 ◽  
Vol 03 (04) ◽  
pp. 463-476 ◽  
Author(s):  
HITOSHI NAKADA ◽  
RIE NATSUI
Keyword(s):  
Invariant Measure ◽  
Continued Fractions ◽  
The Other ◽  
Strong Mixing ◽  
Gauss Measure ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Mixing Properties ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

It is well known that the sequence of digits of regular continued fractions with the Gauss measure is ψ-mixing. In this paper we consider a class of semiregular continued fractions which are called α-continued fractions for α, 1/2≤α≤1. We show that the sequence of digits of α-continued fractions with the absolutely continuous invariant measure is absolutely regular for every α, 1/2≤α≤1, on the other hand, it is not φ-mixing for almost every α, 1/2≤α≤1.

Backward continued fractions, Hecke groups and invariant measures for transformations of the interval

1996 ◽  
Vol 16 (6) ◽  
pp. 1241-1274 ◽  
Author(s):  
Karlheinz Gröchenig ◽  
Andrew Haas
Keyword(s):  
Riemann Surface ◽  
Invariant Measure ◽  
Symbolic Dynamics ◽  
Continued Fractions ◽  
Geodesic Flow ◽  
Invariant Measures ◽  
Interval Maps ◽  
Hecke Groups ◽  
Type Group ◽  
New Type

AbstractWe develop a new type of backward continued fractions that can be associated to each Hecke-type group. We study its symbolic dynamics, and the corresponding interval maps and their invariant measures. These measures are infinite if and only if the corresponding groups are discrete. For the discrete Hecke groups the invariant measure is computed explicitly by studying the geodesic flow on the associated Riemann surface.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

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