A map $f: X \to X$ is topologically transitive if for any pair of non-empty open sets $U, V \subset X$, there exists $n > 0$ such that $f^n(U) \cap V \neq \emptyset$. The system cannot be decomposed into disjoint invariant subsystems; trajectories from any region eventually visit every other region.

A stronger property is topological mixing: for any open sets $U, V$, there exists $N$ such that $f^n(U) \cap V \neq \emptyset$ for all $n > N$. Mixing implies that the system thoroughly shuffles points, continually redistributing them throughout the attractor.

Topological entropy $h_{top}(f)$ measures the exponential growth rate of distinguishable orbits. Roughly, it quantifies how many $\epsilon$-separated orbits can be distinguished over time intervals of length $n$ as $n \to \infty$:

$h_{top}(f) = \lim_{\epsilon \to 0} \limsup_{n \to \infty} \frac{1}{n} \log N(n, \epsilon)$

where $N(n, \epsilon)$ is the maximum number of points with distinct $n$-step orbit histories when measured to precision $\epsilon$.

• $h_{top} = 0$: simple dynamics (periodic, quasi-periodic)
• $h_{top} > 0$: complex dynamics, often chaotic

Strange attractors often have non-integer fractal dimension, reflecting self-similar structure at multiple scales. Common dimension measures include:

• Box-counting dimension: $d_B = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$ where $N(\epsilon)$ is the number of boxes of size $\epsilon$ needed to cover the attractor

• Correlation dimension: based on pairwise distances of points on the attractor

• Lyapunov dimension: relates to the spectrum of Lyapunov exponents

Fractal dimension between 2 and 3 for the Lorenz attractor reflects its intricate, layered structure.