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Category Archives: theorems
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
Posted in ergodic theory, theorems
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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
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
Tagged central limit theorem, martingales
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The PerronFrobenius 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
Posted in theorems
Tagged Hilbert metric, Markov chains, PerronFrobenius, spectral methods
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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
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
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
The stable manifold theorem
The stable manifold theorem is one of the most important in the theory of nonlinear 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