We outline the construction of Smale's horseshoe map and prove it exhibits chaos by conjugacy to the shift map.

Step 1: Geometric Setup

Consider a unit square $D = [0,1] \times [0,1]$. The horseshoe map $f: D \to \mathbb{R}^2$ performs:

1. Stretching: Stretch $D$ vertically by a factor $>2$ and compress horizontally by a factor $<1/2$
2. Folding: Fold the stretched rectangle into a horseshoe shape
3. Placement: Position the horseshoe so it intersects $D$ in two vertical strips

The image $f(D)$ consists of two vertical strips $V_0$ and $V_1$ within $D$. Let $H_0$ and $H_1$ be the two horizontal strips that map to $V_0$ and $V_1$ respectively.

Step 2: Invariant Set

The invariant set is:

$\Lambda = \bigcap_{n=-\infty}^{\infty} f^n(D)$

A point $x \in \Lambda$ if and only if its forward and backward orbits remain in $D$ for all time. Since $f^n(D)$ consists of $2^n$ vertical strips (each of width $\approx (1/2)^n$) and $f^{-n}(D)$ consists of $2^n$ horizontal strips, $\Lambda$ is the intersection of infinitely many stripe patterns, forming a Cantor set structure: uncountable, totally disconnected, nowhere dense.

Step 3: Symbolic Dynamics

Define a symbolic itinerary for each $x \in \Lambda$ by the sequence $s(x) = (\ldots, s_{-1}, s_0, s_1, \ldots)$ where:

$s_n = \begin{cases} 0 & \text{if } f^n(x) \in H_0 \\ 1 & \text{if } f^n(x) \in H_1 \end{cases}$

This defines a map $\phi: \Lambda \to \Sigma_2$, where $\Sigma_2 = \{0,1\}^{\mathbb{Z}}$ is the space of bi-infinite binary sequences with the shift map $\sigma: \Sigma_2 \to \Sigma_2$ defined by $(\sigma(s))_n = s_{n+1}$.

Step 4: Conjugacy

The key is to prove that $\phi$ is a homeomorphism satisfying $\phi \circ f = \sigma \circ \phi$, i.e., $f$ on $\Lambda$ is conjugate to the shift:

• Surjectivity: For any sequence $s \in \Sigma_2$, there exists a unique point $x \in \Lambda$ with itinerary $s$. This follows from the nested intersection property: $\bigcap_{n} H_{s_n}$ for the backward orbit and $\bigcap_{n} V_{s_n}$ for the forward orbit intersect in exactly one point.

• Injectivity: Different sequences correspond to different points because the strips separate the space.

• Continuity: The map $\phi$ is continuous because nearby points have similar (long common) itineraries.

• Conjugacy: By construction, $f(x) \in H_j$ if and only if $x \in H_j$ gets mapped to $V_j$, which corresponds to shifting the sequence: $s_n(f(x)) = s_{n+1}(x)$, i.e., $\phi(f(x)) = \sigma(\phi(x))$.

Step 5: Chaotic Properties

Since the shift map $\sigma$ is known to be chaotic (sensitive dependence, transitivity, dense periodic points), and $f|\Lambda$ is conjugate to $\sigma$, it inherits all these properties:

• Sensitive dependence: Sequences differing in the 0-th position correspond to points separated by a definite distance
• Transitivity: Any finite block appears in any other sequence eventually
• Dense periodic points: Periodic sequences (e.g., $\overline{01}$) are dense

Conclusion: The horseshoe map exhibits deterministic chaos with a fractal invariant set having Cantor structure, proving that stretching and folding are sufficient mechanisms for generating complex dynamics.