# Category Archives: theorems

Important theorems, usually “classical” and “fundamental” in some sense

## Specification and the measure of maximal entropy

These are notes for a talk I am giving in Jon Chaika’s online working seminar in ergodic theory. The purpose of the talk is to outline Bowen’s proof of uniqueness of the measure of maximal entropy for shift spaces with … Continue reading

## Law of large numbers for dependent but uncorrelated random variables

One of the fundamental results in probability theory is the strong law of large numbers, which was discussed in an earlier post under the guise of the Birkhoff ergodic theorem. Suppose we have a sequence of random variables which take … Continue reading

Posted in statistical laws, theorems | Tagged | 4 Comments

## Sharkovsky’s theorem

This post is based on a talk given by Keith Burns in the UH dynamical systems seminar yesterday, in which he presented a streamlined proof of Sharkovsky’s theorem due to him and Boris Hasselblatt. 1. Background and statement of the … Continue reading

## Central Limit Theorem for dynamical systems using martingales

This post is based on notes from Matt Nicol’s talk at the UH summer school in dynamical systems. The goal is to present the ideas behind a proof of the central limit theorem for dynamical systems using martingale approximations. 1. … Continue reading

Posted in ergodic theory, statistical laws, theorems | | 4 Comments

## The Perron-Frobenius theorem and the Hilbert metric

In the last post, we introduced basic properties of convex cones and the Hilbert metric. In this post, we look at how these tools can be used to obtain an explicit estimate on the rate of convergence in the Perron–Frobenius … Continue reading

## Spectral methods 3 – central limit theorem

With the previous post on convergence of random variables, the law of large numbers, and Birkhoff’s ergodic theorem as background, we return to the spectral methods discussed in the first two posts in this series. This post is based on … Continue reading

Posted in statistical laws, theorems | | 8 Comments

## Laws of large numbers and Birkhoff’s ergodic theorem

In preparation for the next post on the central limit theorem, it’s worth recalling the fundamental results on convergence of the average of a sequence of random variables: the law of large numbers (both weak and strong), and its strengthening … Continue reading

Posted in ergodic theory, statistical laws, theorems | Tagged | 7 Comments

## Function spaces and compactness

In the last two posts on spectral methods in dynamics, we’ve used (both explicitly and implicitly) a number of results and a good deal of intuition on function spaces. It seems worth discussing these a little more at length, as … Continue reading

Posted in functional analysis, theorems | | 3 Comments

## The stable manifold theorem

The stable manifold theorem is one of the most important in the theory of non-linear ODEs and dynamical systems. Unfortunately, some of the standard introductory texts (Hirsch–Smale, Perko) either do not give a proof, or do not motivate the proof, … Continue reading

Posted in ODE, smooth dynamics, theorems | | 3 Comments