The Bernoulli shift on $\Sigma_k = \{0, 1, \ldots, k-1\}^{\mathbb{Z}}$ with product measure $\mu = \prod_{n=-\infty}^\infty \nu$ (where $\nu$ is a probability vector on $\{0, \ldots, k-1\}$) and shift map $\sigma$ is:

• Measure-preserving (by product structure)
• Ergodic (shift is topologically transitive)
• Mixing of all orders (independence of coordinates)
• KS entropy: $h_\mu(\sigma) = -\sum_{i=0}^{k-1} \nu_i \log \nu_i$ (Shannon entropy of $\nu$)

For uniform $\nu_i = 1/k$, entropy is $\log k$. Bernoulli shifts are paradigmatic chaotic systems with maximal randomness.

The doubling map $D(\theta) = 2\theta \pmod{2\pi}$ on $S^1$ with Lebesgue measure is:

• Ergodic (orbits are dense for almost all initial conditions)
• Mixing (correlations decay exponentially)
• KS entropy: $h(D) = \log 2$
• Conjugate to the Bernoulli 2-shift via binary expansion

This demonstrates that simple formulas can generate complex ergodic behavior. The map is chaotic (positive entropy) yet exactly solvable through symbolic dynamics.

The geodesic flow on the unit tangent bundle of a compact surface with negative curvature (e.g., hyperbolic plane modulo a lattice) is:

• Volume-preserving (Liouville measure)
• Ergodic (Hopf's theorem)
• Mixing (Hopf, Anosov)
• Positive entropy proportional to curvature

This geometric example shows ergodic theory applies beyond symbolic or discrete systems to continuous flows arising in geometry and physics.

The cat map $A: T^2 \to T^2$ given by $\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \pmod{1}$ is:

• Measure-preserving (determinant 1)
• Ergodic (eigenvalues are irrational)
• Mixing (hyperbolic toral automorphism)
• KS entropy: $h(A) = \log \lambda_{\max} = \log\left(\frac{3 + \sqrt{5}}{2}\right)$

where $\lambda_{\max}$ is the expanding eigenvalue. This is an Anosov system—hyperbolic dynamics on the torus.

Generic Hamiltonian systems on compact energy surfaces exhibit mixed behavior:

• KAM tori: Quasi-periodic, zero entropy regions
• Chaotic seas: Positive entropy, mixing regions
• Separatrices: Boundaries with complex structure

The relative measures of regular versus chaotic regions depend on parameters. For small perturbations of integrable systems, KAM tori dominate (by measure), but chaotic regions provide ergodic behavior and enable transport.