Quantitative Shadowing and Dynamical Complexity: Two Anosov Counterexamples

We construct two families of Anosov automorphisms on a fixed flat two-dimensional torus to compare topological entropy with the optimal local Lipschitz shadowing constant. In the first family, entropy tends to infinity while the shadowing constant tends to one. In the second, the maps are smoothly conjugate and have identical entropy, periodic-point counts, and eigenvalues, but their shadowing constants diverge. Thus neither entropy nor the full periodic-point counting sequence gives a finite upper bound for the shadowing constant. The constructions distinguish the effects of expansion strength and invariant-direction geometry.