Quantitative Shadowing and Dynamical Complexity: Two Anosov Counterexamples

Can topological entropy predict how accurately approximate orbits can be shadowed by true orbits? We show that it cannot, even for Anosov automorphisms on the same two-dimensional torus with the same flat metric. The optimal local Lipschitz shadowing constant is the infimum of error-amplification factors valid at all sufficiently small scales. We construct one family whose entropy tends to infinity while this constant tends to the optimal lower bound one. We construct another family whose maps are smoothly conjugate and have identical entropy, periodic-point counts, and eigenvalues, while their shadowing constants tend to infinity. Thus neither entropy nor the full periodic-point counting sequence provides a finite upper bound. The proofs compute the toral constant through a linear inverse error operator and distinguish the effects of expansion strength and the angle between stable and unstable directions.