Questions: Symbolic Dynamics

Question 1

In the horseshoe map with two strips labeled 0 and 1, the number of periodic orbits of period n is:

• A. n
• B. 2^n (the number of binary strings of length n, though some are cyclic permutations of others)
• C. n!
• D. Fibonacci(n)

Answer: (B)

Each periodic orbit of period n corresponds to a repeating binary sequence of period n. There are 2^n binary strings of length n, so there are 2^n fixed points of the n-th iterate of the map. The number of prime periodic orbits (orbits with minimal period exactly n) is obtained by excluding sequences whose period is a proper divisor of n — this gives (2^n - Σ_{d|n, d<n} #orbits of period d)/n by Mobius inversion. But the total count of period-n fixed points is exactly 2^n, growing exponentially.

Question 2

The topological entropy of the full shift on two symbols is ln 2. What does this quantity measure physically?

• A. The rate of energy dissipation in the system
• B. The rate at which the system generates information — equivalently, the exponential growth rate of the number of distinguishable orbits as the observation time increases
• C. The temperature of the chaotic attractor
• D. The fractal dimension of the invariant set

Answer: (B)

Topological entropy measures the complexity of the dynamics by counting how fast the number of distinguishable orbit types grows with time. For the full shift on k symbols, the number of distinct sequences of length n is k^n, so the entropy is lim (1/n) ln k^n = ln k. For k = 2, this is ln 2 ≈ 0.693 bits per iteration. This equals the largest Lyapunov exponent for the corresponding expanding map, connecting symbolic dynamics to the quantitative theory of chaos.

Question 3

Symbolic dynamics turns the study of a chaotic map into the study of sequences of symbols. This encoding always preserves all dynamical information exactly.

• T. True
• F. False

Answer: False

For hyperbolic systems (like the horseshoe), the encoding is exact — there is a topological conjugacy between the map on the invariant set and the shift on symbol sequences. But for non-hyperbolic systems, the encoding may lose information: distinct points might have the same symbol sequence (the partition doesn't separate them), or the correspondence might fail to be continuous. The partition must be a Markov partition for the encoding to be exact. Constructing such partitions for general systems is difficult, and for non-uniformly hyperbolic systems, the symbolic dynamics is only an approximation.

Question 4

How does symbolic dynamics make it easy to prove that the horseshoe has sensitive dependence on initial conditions?

Model answer: Two points with different symbol sequences must eventually differ in some position (say position n). This means that after n iterations, the two orbits visit different strips — they are in macroscopically different parts of phase space. Two points that are very close in the Cantor set can differ in an arbitrarily early position of their symbol sequence (because the Cantor set interleaves points with different encodings at every scale). Therefore, orbits that start arbitrarily close can end up in different regions after a finite number of steps — this is sensitive dependence. The proof is essentially combinatorial, requiring no analysis of differential equations.

This is the power of the symbolic approach: deep dynamical properties become trivial combinatorial observations. The shift map on {0,1}^Z is obviously sensitive to initial conditions (changing one symbol in a sequence creates a sequence that diverges after that position), obviously topologically transitive (any finite block appears in some sequence), and obviously has dense periodic orbits (approximate any sequence by a periodic repetition of a long block). Since the horseshoe is conjugate to the shift, it inherits all these properties automatically.