QUASI-SHADOWING PROPERTY FOR NONUNIFORMLY PARTIALLY HYPERBOLIC SYSTEMS

The paper proves a quasi-shadowing theorem for nonuniformly partially hyperbolic systems (Theorem A) using “fake” invariant foliations built in Pesin/Lyapunov charts, product-structure cylinders, and a leaf-intersection construction that produces the tracing points t(zi) and yields uniform bounds along each orbit segment and across center jumps (see the statement of Theorem A and the setup with fake foliations and cylinders, including the invariance and Hölder regularity properties, and the key distance estimates along segments and across jumps ). The candidate solution reaches the same conclusion but via a different, standard contraction scheme: it composes the stable contraction along each orbit segment with a center projection across the small jump, obtaining uniformly contracting transition maps Tn on stable plaques and then takes limits from the past. This differs in presentation from the paper’s leaf-selection method, but it is compatible with the paper’s fake-foliation framework and assumptions. The main caveat is that the model implicitly assumes Lipschitz control for the center holonomy/projection Πn on stable plaques; the paper provides the requisite local-product Lipschitz control (constant b close to 1 after choosing parameters) and exponential bounds, which suffice to validate the contraction argument if stated carefully (see the local-product inequalities and the choice of parameters ensuring a contraction factor below 1 ).