The quadratic family $f_c(z) = z^2 + c$ with $c \in \mathbb{C}$ is central to complex dynamics. For each parameter $c$, the filled Julia set $K_c$ consists of points with bounded orbits:

$K_c = \{z \in \mathbb{C} : \{f_c^n(z)\}_{n=0}^\infty \text{ is bounded}\}$

The Mandelbrot set $M$ is defined as the set of parameters for which $0 \in K_c$:

$M = \{c \in \mathbb{C} : f_c^n(0) \not\to \infty\}$

The boundary of $M$ is a fractal with infinite complexity, exhibiting self-similarity at all scales.

A circle map is a continuous map $f: S^1 \to S^1$ of the circle to itself, often written as $f: \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$. The Arnold standard circle map is:

$\theta_{n+1} = \theta_n + \Omega - \frac{K}{2\pi}\sin(2\pi\theta_n) \pmod{1}$

where $\Omega$ is the bare rotation number and $K$ measures nonlinearity. For $K = 0$, this reduces to rigid rotation. For $K > 0$, the map exhibits mode locking where the rotation number becomes rational on open intervals (Arnold tongues).

The doubling map (or binary shift) $\sigma(x) = 2x \pmod{1}$ on $[0,1)$ demonstrates several important concepts:

• Every dyadic rational is periodic
• Non-dyadic rationals have dense orbits
• The map is chaotic with positive Lyapunov exponent $\ln 2$
• It is conjugate to the left shift on binary sequences

The doubling map connects geometric iteration with symbolic dynamics through binary expansions.

The Gauss map $G(x) = \frac{1}{x} \pmod{1}$ on $(0,1]$ is intimately related to continued fractions. For $x = [a_1, a_2, a_3, \ldots]$ in continued fraction notation:

$G^n(x) = [a_{n+1}, a_{n+2}, a_{n+3}, \ldots]$

This map has remarkable ergodic properties with invariant measure:

$d\mu = \frac{1}{\ln 2} \cdot \frac{dx}{1+x}$

The Gauss map provides a dynamical perspective on number-theoretic properties of continued fractions.