Ergodic Theory

Ergodic Theory (Winter 2024), at least in the context we will be studying it, is about properties that hold for the vast majority of real numbers.

A typical example is the question of normality of numbers. If we take a typical real number, then its decimal expansion will have roughly the same number of 0’s, 1’s, 2’s, and so forth. Similarly, it will have roughly the same number of 00’s, 01’s, 38’s, and so forth appearing in two-digit windows. By a “typical” number, we mean that most numbers have this property. A key step here is to define exactly what it means for “most” numbers to have some property, given that infinitely many (and even uncountably many) do not. We will make this precise with the introduction of measure theory. Then we shall prove that most numbers are normal in the above sense. Similarly, we shall prove Khinchin’s Theorem, which says that there is some specific number, known as Khinchin’s constant, such that the geometric mean of the coefficients in the continued fraction of most numbers converges to Khinchin’s constant.

A curious aspect of these ergodic theory-style theorems is that, while they tell us properties of almost all numbers, it is still extremely difficult to say anything about any specific number. For instance, everyone believes that \pi is normal and satisfies Khinchin’s Theorem, but no one has any idea of how to prove it.

Along the way, we may take various detours to introduce the modern theory of integration, and also how to formalize probability using measure theory, depending on time.

Calculus will be used occasionally in this class, but students who are not familiar with calculus will still be able to follow.