Shadowing and Generalized Hyperbolicity for Dissipative Composition Operators without Bounded Distortion

Let T_f φ = φ∘f be a dissipative composition operator on L^p(X, B, μ), 1 ≤ p < ∞, induced by a bimeasurable bijection f such that μ∘f and μ∘f⁻¹ are bounded by a multiple of μ. D'Aniello, Darji and Maiuriello proved that if f has bounded distortion, then T_f has the shadowing property if and only if it is generalized hyperbolic, if and only if the masses μ(f^k(W)) of the iterates of a wandering set satisfy one of three growth conditions. They asked, and Maiuriello asked again in Oberwolfach Report 19/2024, whether this remains true without bounded distortion. We show that shadowing and generalized hyperbolicity are equivalent for every dissipative composition operator, for all 1 ≤ p < ∞, for real or complex scalars and without separability assumptions. Both are equivalent to the surjectivity of I − T_f, and to a uniform exponential "tent" condition on the Radon–Nikodym densities d(μ∘f^k)/dμ on the wandering set. The proof represents T_f as a field of weighted shifts and rests on a deterministic lemma about a single fibre. For p = 2, complex scalars and separable L²(μ) the equivalence also follows from a recent theorem of Pituk on separable Hilbert spaces. The characterization by masses fails without bounded distortion in both directions: Bernardes, D'Aniello and Maiuriello recently gave an example satisfying the contraction condition (HC) without shadowing, and we record an elementary hyperbolic example that satisfies none of the three conditions. This is an unrefereed note.