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A Borel-Cantelli lemma for non-stationary dynamical systems

Åberg, Teodor LU (2023) In Master'sTheses in Mathematical Sciences FMAM05 20231
Mathematics (Faculty of Engineering)
Abstract
In this thesis we present a Borel–Cantelli lemma for dynamical systems
such that the dynamics driving the system varies with time. These results
are largely based on a decomposition of transfer operators produced by
Rychlik. Some limitations on the structure of sequences that is necessary
for the results is given some discussion. Two examples of systems that
could be considered are presented, one of which is accompanied by a
method by Liverani for bounding rates of decay.
Popular Abstract
The function is one of the most fundamental, and recognizable, objects of
study in mathematics. Being as widespread as they are the amount of questions
that one could ask about their behaviour seems endless. If one has a function
f, such that the output y = f (x) is a valid input to the function one could
examine what happens if one applies the function to its output, over and over
again.
In the study of discrete dynamical systems questions of this sort are considered. What one can say about the evolution of the system varies drastically
depending on what function is used. For instance one can find functions where
the iterates approach a single point, no matter where one starts. On the other
hand, one can also find functions... (More)
The function is one of the most fundamental, and recognizable, objects of
study in mathematics. Being as widespread as they are the amount of questions
that one could ask about their behaviour seems endless. If one has a function
f, such that the output y = f (x) is a valid input to the function one could
examine what happens if one applies the function to its output, over and over
again.
In the study of discrete dynamical systems questions of this sort are considered. What one can say about the evolution of the system varies drastically
depending on what function is used. For instance one can find functions where
the iterates approach a single point, no matter where one starts. On the other
hand, one can also find functions where, even for points that are initially very
close to one another, behaviour can vary wildly.
This thesis handles systems that fall into the second category. For these
types of systems it is difficult to make statements for the iterates starting at
some specific point. Instead one can try to find properties that are fulfilled very
commonly. The property that is examined here is the so called Borel–Cantelli
property, which can be understood as finding conditions such that one can say
something about the value of
f (. . . f (x) . . . ) n times
for infinitely many different values of n and for basically any initial point x.
Some known results of this type are presented, as well as a new result where,
instead of a single function used when iterating, functions from a given family
are used instead. Some adjacent previous results are also mentioned, as they
are useful when examining the conditions that are necessary for the new result
to be valid. (Less)
Please use this url to cite or link to this publication:
author
Åberg, Teodor LU
supervisor
organization
course
FMAM05 20231
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Borel-Cantelli lemmas, Dynamical systems
publication/series
Master'sTheses in Mathematical Sciences
report number
LUTFMA-3515-2023
ISSN
1404-6342
other publication id
2023:E46
language
English
id
9127787
date added to LUP
2023-06-21 08:22:30
date last changed
2023-06-21 08:22:30
@misc{9127787,
  abstract     = {{In this thesis we present a Borel–Cantelli lemma for dynamical systems
such that the dynamics driving the system varies with time. These results
are largely based on a decomposition of transfer operators produced by
Rychlik. Some limitations on the structure of sequences that is necessary
for the results is given some discussion. Two examples of systems that
could be considered are presented, one of which is accompanied by a
method by Liverani for bounding rates of decay.}},
  author       = {{Åberg, Teodor}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master'sTheses in Mathematical Sciences}},
  title        = {{A Borel-Cantelli lemma for non-stationary dynamical systems}},
  year         = {{2023}},
}