(d) Fix a point p. For q ∈ W^{u}(p) \ {p}, the distance d(f_{A}^{k}(p), f_{A}^{k}(q)) ≥ λ^{k}d(p, q) as long as the distance stays less than one half. The quantity λ^{k}d(p, q) grows as k > 0 increases, so this distance gets bigger than 1/(2λ) for some k > 0. This proves that f_{A} has sensitive dependence. Similarly, for q ∈ W^{s}(p) \ {p}, the distance d(f_{A}^{k}(p), f_{A}^{k}(q)) ≥ μ^{k}d(p, q) grows as k < 0 becomes more negative, so it becomes bigger than μ/2 for some k < 0.

Finally, there is an δ_{0} > 0 such that for any point q ≠ p with d(p, q) < δ_{0}, the two points q_{u} ∈ W^{u}(p)∩W^{s}(q) and q_{s} ∈ W^{s}(p)∩W^{u}(q) are within 1/(2λ^{2}) and μ^{2}/2 respectively of p. As least one of these two points is distinct from p; let us take the case when d(p, q_{u}) ≥ d(p, q_{s}). Because the stable and unstable manifolds are parallel lines in the universal covering space, d(p, q_{u}) = d(q_{s}, q) and d(p, q_{s}) = d(q_{u}, q). There is a first k_{1} ≥ 1 such that d(f^{k_{1}}(p), f^{k_{1}}(q_{u})) ≥ 1/(2λ). Then

d(f^{k_{1}}(p), f^{k_{1}}(q)) ≥ d(f^{k_{1}}(p), f^{k_{1}}(q_{u})) − d(f^{k_{1}}(q_{u}), f^{k_{1}}(q))

≥ \frac{1}{2λ} − μ d(q_{u}, q)

≥ \frac{1}{2λ} − μ d(p, q_{s})

≥ \frac{1}{2λ} − μ (\frac{1}{2λ^{2}} )

= (1 − \frac{μ}{λ})(\frac{1}{2λ}).

Similarly, if d(p, q_{s}) ≥ d(p, q_{u}), then for some k_{1} < 0

d(f^{k_{1}}(p), f^{k_{1}}(q)) ≥ (1 − \frac{μ}{λ})(\frac{μ}{2}).

Therefore, f_{A} is expansive with expansive constant

min \{δ_{0}, (1 − \frac{μ}{λ})\frac{1}{2λ}, (1 − \frac{μ}{λ})\frac{μ}{2}\}

This proves part (d).