This course develops the modern theory of deterministic chaos and ergodic behavior in dynamical systems. It asks how simple nonlinear rules can produce complicated long-term behavior, and how that behavior can still be described using geometry, symbolic coding, and statistical laws. The central objects are recurrent orbits, invariant sets, hyperbolic dynamics, and invariant measures, with an emphasis on understanding when chaos is robust, when it can be classified, and how it can be measured.

The chapters begin with recurrence and the first signs of chaotic motion, then introduce symbolic dynamics as a way to encode orbits by sequences. From there the course builds the geometric theory of horseshoes, Smale dynamics, hyperbolic sets, stable manifolds, and homoclinic intersections, leading to shadowing and structural stability. Later chapters turn to quantitative complexity through topological entropy, then to invariant measures, ergodicity, Lyapunov exponents, SRB measures, and physical measures. The final chapters bring these strands together through Markov partitions and thermodynamic formalism, showing how geometry, coding, and statistics form a unified picture of chaotic dynamics.

# Introduction

This introductory chapter fixes the language and expectations for the course. The main problem is to understand how deterministic rules can produce behaviour that is geometrically organized, topologically complicated, and statistically regular. We will move between maps, flows, invariant sets, symbolic codings, entropy, Lyapunov exponents, and invariant measures, so the first task is to identify the common objects that all later chapters will refine.

The course assumes a first encounter with dynamical systems: fixed points, invariant manifolds, elementary stability, and examples from ordinary differential equations. Here the emphasis shifts from local phase portraits to long-term behaviour on invariant sets. A recurring theme is that chaos is not a synonym for disorder; it is the presence of rigid structures that make complicated orbit patterns analyzable.

## The Central Question of Chaotic Dynamics

What does it mean to understand a system after individual orbit prediction has become unreliable? For a map $f:X \to X$ or a flow $(\varphi_t)_{t \in \mathbb R}$ on a phase space $X$, the raw data are orbits, but the objects we can often control are invariant sets, recurrence properties, rates of separation, and invariant probability measures. The course develops these objects as complementary ways of describing the same dynamics.

[definition: Discrete-Time Dynamical System] A discrete-time dynamical system is a pair $(X,f)$ where $X$ is a phase space and $f:X \to X$ is a map. The forward orbit of $x \in X$ is the sequence $(f^n(x))_{n \ge 0}$, where $f^0=\operatorname{id}_X$ and $f^{n+1}=f \circ f^n$. [/definition]

This definition records the basic mechanism for maps: the system evolves by repeated composition. Many systems in applications evolve continuously in time rather than by a fixed update rule, so we also need notation for families of maps whose parameter is time itself.

[definition: Continuous-Time Dynamical System] A continuous-time dynamical system is a family $(\varphi_t)_{t \in \mathbb R}$ of maps $\varphi_t:X \to X$ satisfying $\varphi_0=\operatorname{id}_X$ and $\varphi_{t+s}=\varphi_t \circ \varphi_s$ for all $s,t \in \mathbb R$. The orbit of $x \in X$ is the set $\{\varphi(t,x):t \in \mathbb R\}$. [/definition]

The same data can be packaged as a single action $\varphi:\mathbb R \times X \to X$, written $\varphi(t,x)=\varphi_t(x)$; the group law $\varphi_{t+s}=\varphi_t \circ \varphi_s$ then becomes a joint condition on the two-variable map. Flows enter because many systems arise from differential equations, while maps arise both directly and as time-one maps or return maps. The bridge between them lets us transfer ideas such as recurrence, invariant measures, and entropy across continuous and discrete time.

[example: Time-One Map of a Flow] Let $(\varphi_t)_{t \in \mathbb R}$ be the flow of an autonomous ordinary differential equation $\dot{x}=F(x)$ on a smooth manifold $M$, so $\varphi_0=\operatorname{id}_M$ and $\varphi_{t+s}=\varphi_t \circ \varphi_s$. Define $f=\varphi_1:M \to M$. Then the first few iterates are \begin{align*} f^0=\operatorname{id}_M=\varphi_0. \end{align*} Also, \begin{align*} f^1=f=\varphi_1. \end{align*} For the second iterate, the flow law gives \begin{align*} f^2=f\circ f=\varphi_1\circ \varphi_1=\varphi_{1+1}=\varphi_2. \end{align*} If $f^n=\varphi_n$, then \begin{align*} f^{n+1}=f\circ f^n=\varphi_1\circ \varphi_n=\varphi_{1+n}=\varphi_{n+1}. \end{align*} Thus, by induction, $f^n=\varphi_n$ for every $n\ge 0$. For each $x\in M$, the forward orbit of the discrete-time system $(M,f)$ is therefore \begin{align*} \mathcal O_f^+(x)=\{f^n(x):n\ge 0\}=\{\varphi_n(x):n\ge 0\}. \end{align*} This is exactly the set of points obtained by sampling the continuous trajectory $t\mapsto \varphi_t(x)$ at integer times. The time-one map preserves the integer-time skeleton of the flow, while behaviour on intervals such as $0<t<1$ is invisible unless we keep the full family $(\varphi_t)_{t\in\mathbb R}$. [/example]

The example shows that different time descriptions can point to the same long-term motion. To compare such descriptions, we need to isolate the subsets that keep the dynamics inside themselves and therefore carry their own internal orbit structure.

[definition: Invariant Set] Let $(X,f)$ be a discrete-time dynamical system. A subset $A \subset X$ is forward invariant if $f(A) \subset A$, invariant if $f(A)=A$, and completely invariant if $f^{-1}(A)=A$. [/definition]

For non-invertible maps, these distinctions matter. Forward invariance is enough for future evolution to remain in $A$, while stronger versions keep track of preimages and are often needed when coding all bi-infinite orbit histories.

[example: Invariance for the Doubling Map] View $S^1$ as $\mathbb R/\mathbb Z$, so $D([x])=[2x]$. The whole circle is invariant because for every $[y]\in S^1$, the point $[y/2]\in S^1$ satisfies \begin{align*} D([y/2])=[2(y/2)]=[y]. \end{align*} Thus $D(S^1)=S^1$. If $x$ is periodic with period $p$, so $D^p(x)=x$, its periodic orbit is \begin{align*} P=\{x,D(x),D^2(x),\ldots,D^{p-1}(x)\}. \end{align*} For $0\le k\le p-2$, applying $D$ to $D^k(x)$ gives $D^{k+1}(x)\in P$, and for the last point, \begin{align*} D(D^{p-1}(x))=D^p(x)=x\in P. \end{align*} Hence $D(P)\subset P$. Since every element of $P$ is also the image under $D$ of the preceding element in the same cycle, $D(P)=P$. The symbolic picture comes from binary expansions. If \begin{align*} x=0.b_1b_2b_3\cdots=\sum_{j\ge 1} b_j2^{-j}, \end{align*} with $b_j\in\{0,1\}$, then \begin{align*} 2x=b_1+\sum_{j\ge 1}b_{j+1}2^{-j}. \end{align*} Reducing modulo $1$ removes the integer part $b_1$, so \begin{align*} D(x)=0.b_2b_3b_4\cdots. \end{align*} Thus $D$ acts on binary expansions by shifting the sequence one step to the left. This explains the invariance issue for forbidden words. If a set is defined by forbidding a word everywhere in the binary sequence, then shifting left cannot create a new occurrence of that word, because every block in $b_2b_3b_4\cdots$ was already a block in $b_1b_2b_3\cdots$. That set is forward invariant. If the condition only forbids a word at a fixed initial position, forward invariance can fail: the sequence $0111\cdots$ does not begin with $11$, but after applying $D$ it becomes $111\cdots$, which does begin with $11$. The doubling map therefore turns a geometric question about subsets of the circle into a combinatorial question about which binary sequence conditions are preserved by the left shift. [/example]

## Three Viewpoints on Chaos

How can a single system be described by topology, geometry, and probability at the same time? The same orbit structure may be examined through open sets, through derivatives and stretching rates, or through the distribution of time spent in different regions. The course deliberately keeps these viewpoints in contact.

[definition: Topological Dynamical System] A topological dynamical system is a pair $(X,f)$ where $X$ is a [topological space](/page/Topological%20Space) and $f:X \to X$ is continuous. [/definition]

Continuity makes qualitative orbit behaviour visible through open sets. This topological language is not enough for questions about stretching rates, stable directions, or tangent-vector growth, because open sets do not record how nearby orbits separate infinitesimally. To ask whether a map expands one tangent direction while contracting another, the phase space must carry smooth coordinates and the time evolution must respect that smooth structure.