Doubling map: x → 2x mod 1. The n-th iterate reads off the n-th binary digit of x₀. Sensitive dependence = reading bits you don't know yet. Entropy = log₂(k) bits/step.
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Doubling map orbit (binary expansion)
Uncertainty growth (sensitive dependence)
The Bernoulli shift σ: {0,...,k-1}^ℤ → {0,...,k-1}^ℤ sends (sₙ) → (sₙ₊₁). Topological entropy = log(k). KS metric entropy = Σp_i log(p_i) for the Bernoulli measure. Every orbit is determined by its infinite past — but any finite truncation is unpredictable in the future.