For all x > 0,
exp(f(x)) = g(exp(x)). Orbits of f on the left panel project through y = exp(x) onto orbits of g on the right panel, step by step. Fixed points: x* ≈ 0.34416, y* = exp(x*) ≈ 1.41081, multiplier at each ≈ 4.31642.⟶
exp
step 0/0f-orbit: —g-orbit: —|exp(f) − g|: —
x₀ = 1.000
The EAL and EXL self-maps have different fixed points (0.344 vs 1.411) yet share the same multiplier to 13 decimals (4.3164206…). That's conjugacy at work: dynamical invariants live on conjugacy classes, not individual maps. The cobweb on the left is a 1-to-1 image of the cobweb on the right under exp — whatever happens to the orbit in f-space happens to the orbit in g-space, at the matching point.
Lean formalization:
Lean formalization:
eal_exl_conjugacy in SelfMapConjugacy.lean (0 sorries).