The Smale horseshoe map has a natural Markov partition into two horizontal strips $H_0, H_1$. A point's symbolic itinerary $(... s_{-1} s_0 s_1 ...)$ records which strip contains each iterate. The invariant set $\Lambda$ is conjugate to the full 2-shift $\Sigma_2$ via this coding.

Properties transfer directly:

• Period-$n$ orbits in horseshoe $\leftrightarrow$ period-$n$ sequences in $\Sigma_2$
• Number of period-$n$ orbits: $2^n$
• Topological entropy: $h_{top} = \log 2$

The horseshoe's fractal Cantor structure reflects the shift's discrete combinatorial structure: uncountable points organized by infinite symbolic sequences.

The tent map $T_2(x) = 2\min(x, 1-x)$ on $[0,1]$ partitions as $[0, 1/2)$ and $[1/2, 1]$ (symbols 0 and 1). The coding via base-2 expansion conjugates $T_2$ to the 2-shift:

$x = 0.s_0 s_1 s_2 \ldots \quad \leftrightarrow \quad s = (s_0, s_1, s_2, \ldots)$

Properties:

• Dyadic rationals $\leftrightarrow$ eventually periodic sequences
• Dense orbit $\leftrightarrow$ dense sequence (e.g., $0.0110010110011 \ldots$ from Thue-Morse)
• Sensitive dependence: differing in $s_n$ means distance $\approx 2^{-n}$ grows to $\approx 1/2$

This explicit conjugacy makes the tent map's chaos rigorously analyzable.

A sofic shift is the image of an SFT under a finite-to-one factor map. Equivalently, it's the set of bi-infinite words generated by a finite automaton. Example: the even shift consists of sequences with even gaps between 1's:

$\Sigma_{\text{even}} = \{\ldots 010010100100 \ldots, \ldots 000100010000 \ldots, \ldots\}$

This is sofic but not SFT. Sofic shifts generalize SFTs, capturing broader classes of constraints while remaining tractable through automata theory.

For $\beta > 1$, the $\beta$-transformation $T_\beta(x) = \beta x \pmod{1}$ generalizes the doubling map. The symbolic dynamics uses base-$\beta$ expansions. For $\beta = (1+\sqrt{5})/2$ (golden ratio), admissible sequences satisfy the golden mean constraint (no consecutive 1's).

Beta-shifts connect to:

• Algebraic number theory (beta as Pisot number)
• Quasi-crystals and aperiodic tilings
• Fractal dimensions of attractors

The interplay between $\beta$'s algebraic properties and symbolic constraints reveals deep connections between dynamics and number theory.