Decay of Correlations for the Gauss Map
(2024) In Master's Theses in Mathematical Sciences FMAM05 20241Mathematics (Faculty of Engineering)
- Abstract
- We study the Gauss map and derive certain statistical properties, specifically upper bounds on the decay of correlations. The method is due to Liverani \cite{Main_Liverani}, which in turn is based on foundational work by Birkhoff \cite{LatticeTheory}. One defines a cone of functions, and then considers the induced Hilbert metric associated to the cone. A strict contraction in this metric gives exponential decay of correlations. First functions belonging to $C^1([0,1])$ are considered and the result is extended to $\text{Lip}([0,1])$. Second we focus on functions of bounded total variation on $[0,1]$. A result in metric number theory regarding growth of continued fractions coefficients follows from the exponential decay of correlations for... (More)
- We study the Gauss map and derive certain statistical properties, specifically upper bounds on the decay of correlations. The method is due to Liverani \cite{Main_Liverani}, which in turn is based on foundational work by Birkhoff \cite{LatticeTheory}. One defines a cone of functions, and then considers the induced Hilbert metric associated to the cone. A strict contraction in this metric gives exponential decay of correlations. First functions belonging to $C^1([0,1])$ are considered and the result is extended to $\text{Lip}([0,1])$. Second we focus on functions of bounded total variation on $[0,1]$. A result in metric number theory regarding growth of continued fractions coefficients follows from the exponential decay of correlations for $BV$. The proof makes use of a dynamical Borel--Cantelli lemma. (Less)
- Popular Abstract
- A common way to describe a system that is hard to predict is to call it chaotic. For instance, think about releasing a gas into a closed container. While the position of any individual gas particle isn't truly random, it behaves as if it were, due to the incredible complexity of the system. Understanding how systems like this behave over time is the goal of a field in mathematics called dynamical systems.
In my thesis, "Decay of Correlations for the Gauss Map," I explore an interesting concept in this field. To make it more concrete, I focused on the Gauss map, which is a function that helps generate continued fractions—a special way of breaking down numbers into a sequence of fractions. The Gauss map essentially tells you how to find... (More) - A common way to describe a system that is hard to predict is to call it chaotic. For instance, think about releasing a gas into a closed container. While the position of any individual gas particle isn't truly random, it behaves as if it were, due to the incredible complexity of the system. Understanding how systems like this behave over time is the goal of a field in mathematics called dynamical systems.
In my thesis, "Decay of Correlations for the Gauss Map," I explore an interesting concept in this field. To make it more concrete, I focused on the Gauss map, which is a function that helps generate continued fractions—a special way of breaking down numbers into a sequence of fractions. The Gauss map essentially tells you how to find the next fraction in this sequence.
One key idea I investigate is how patterns in a system fade as time goes on, a process known as "decay of correlations." Imagine knowing what a system looks like now—how much can you predict about its future? Over time, systems tend to "forget" their past behavior, and decay of correlations measures how quickly that forgetting happens.
In the case of the Gauss map, I used mathematical techniques to show that this forgetting process happens quite fast. This is crucial because understanding how quickly a system loses its memory helps us predict its long-term behavior more accurately.
I also looked at different ways of analyzing the system to see how they respond to the Gauss map. This approach allowed me to extend our understanding of how these decay patterns work, providing new insights into how quickly systems like the Gauss map forget their past. My work is a step towards better grasping the complex and often unpredictable nature of dynamical systems, with potential applications in many areas of science and technology. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9171181
- author
- Ekström, Samuel LU
- supervisor
- organization
- course
- FMAM05 20241
- year
- 2024
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Dynamical systems, measure theory, Gauss map, continued fractions, correlation, exponential decay of correlations, decay of correlations, projective metric, Hilbert metric, Borel-Cantelli
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUTFMA-3540-2024
- ISSN
- 1404-6342
- other publication id
- 2024:E32
- language
- English
- id
- 9171181
- date added to LUP
- 2024-08-14 12:37:18
- date last changed
- 2024-08-14 12:37:18
@misc{9171181,
abstract = {{We study the Gauss map and derive certain statistical properties, specifically upper bounds on the decay of correlations. The method is due to Liverani \cite{Main_Liverani}, which in turn is based on foundational work by Birkhoff \cite{LatticeTheory}. One defines a cone of functions, and then considers the induced Hilbert metric associated to the cone. A strict contraction in this metric gives exponential decay of correlations. First functions belonging to $C^1([0,1])$ are considered and the result is extended to $\text{Lip}([0,1])$. Second we focus on functions of bounded total variation on $[0,1]$. A result in metric number theory regarding growth of continued fractions coefficients follows from the exponential decay of correlations for $BV$. The proof makes use of a dynamical Borel--Cantelli lemma.}},
author = {{Ekström, Samuel}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{Decay of Correlations for the Gauss Map}},
year = {{2024}},
}