Ergodic theory PDF
Preview Ergodic theory
This book is about probabilistic aspects of discrete dynamical systems and
their computer simulations. Basic measure theory is used as a theoretical tool
to describe probabilistic phenomena. All of the simulations are done using the
mathematical software Maple, which enables us to translate theoretical ideas
into computer programs word by word avoiding any other technical details.
The level of familiarity with computer programming is kept minimal. Thus
even theoretical mathematicians can appreciate the experimental nature of
one of the fascinating areas in mathematics.
A new philosophy is introduced even for those who are already familiar
with numerical simulations of dynamical systems. Most computer experiments
in this book employ many significant digits depending on the entropy of the
given transformation so that they can still produce meaningful results after
many iterations. This is why we can do experiments that have not been tried
elsewhere before. It is the main point of the book. Readers should regard the
Maple programs as important as the theoretical explanations of mathematical
facts. The book is designed in such a way that theoretical and experimental
parts complement each other.
A dynamical system is a set of points together with a transformation
rule that describes the movement of points. Sometimes the points lose the
geometrical implication and they are just elements in an abstract set that has
certain structures depending on the type of the problem under consideration.
The theory of dynamical systems studies the long term statistical behavior of
the points under the transformation rules. When the given set of points has
a probabilistic structure, the theory of dynamical systems is called ergodic
theory. The aim of the book is to study computational aspects of ergodic
theory including data compression.
Many mathematical concepts introduced in the book are illustrated by
computer simulations. That gives a different flavor to ergodic theory, and
may attract a new generation of students who like to play with computers.
Some theorems look different, at least to beginners, when the abstract facts
are explained in terms of computer simulations. Many computer experiments
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