*Extracting dynamical behaviour via Markov models*

## Abstract

Statistical properties of chaotic dynamical systems are difficult to estimate reliably. Using long trajectories as data sets sometimes produces misleading results.
It has been recognised for some time that statistical properties are often stable under the addition of a small amount of noise.
Rather than analysing the dynamical system directly, we slightly perturb it to create a Markov model. The analogous statistical properties of the Markov model often have ``closed forms'' and are easily computed numerically. The Markov construction is observed to provide extremely robust estimates and has the theoretical advantage of allowing one to prove convergence in the ``noise approaches zero'' limit and produce rigorous error bounds for quantities.
We review the latest results and techniques in this area.

## Contents

### Chapter 1: Introduction and basic constructions

#### 1.1 What do we do?

#### 1.2: How do we do this?

#### 1.3: Why do we do this?

### Chapter 2: Objects and behaviour of interest

#### 2.1: Invariant measures

#### 2.2: Invariant sets

#### 2.3: Decay of correlations

#### 2.4: Lyapunov exponents

#### 2.5: Mean and variance of return times

### Chapter 3: Deterministic systems

#### 3.1: Basic Constructions

#### 3.2: Invariant measures and invariant sets

#### 3.3: Decay of correlations and spectral approximation

#### 3.4: Lyapunov exponents and entropy

#### 3.5: Mean and variance of return times

### Chapter 4: Random systems

#### 3.1: Basic Constructions

#### 3.2: Invariant measures

#### 3.3: Lyapunov exponents

#### 3.4: Mean and variance of return times

#### 3.5: Advantages for Markov modelling of random dynamical systems

### Chapter 5: Miscellany

#### 5.1: Global attractors, noisy systems, rotation numbers, topological entropy, spectra of ``averaged'' transfer operators for random systems

### Chapter 6: Numerical Tips and Tricks

#### 6.1: Transition matrix construction

#### 6.2: Partition selection