Chaotic weighted translations and orbit-generated ReLU approximation on the noncompact line

Chaos in infinite-dimensional linear systems can arise from a simple combination of transport and spatially heterogeneous amplification. We study this mechanism on 𝐶 0 ⁡ ( ℝ ) , the Banach space of continuous localized profiles, and connect it to finite ReLU approximation of selected orbit states. For a positive weight that contracts the far-left tail and expands the far-right tail, we prove that the weighted translation 𝑇 𝑤 ⁢ 𝑓 ⁡ ( 𝑥 ) = 𝑤 ⁡ ( 𝑥 ) ⁢ 𝑓 ⁡ ( 𝑥 + 1 ) is hypercyclic and has dense periodic points; hence it is chaotic in the standard sense of linear dynamics. The same transport-amplification argument yields chaotic weighted composition operators for increasing, proper, fixed-point-free input deformations, while injectivity, properness, and absence of fixed points are shown to be necessary. A new qualitative obstruction appears when macroscopic endpoint coordinates are retained: their geometric evolution prevents hypercyclicity on the full endpoint spaces. We restore chaotic dynamics on the equal-endpoint space by an explicit rank-one feedback from point evaluation to the endpoint coordinate. On the approximation side, we prove that bounded one-dimensional ReLU networks are uniformly dense in the space of continuous functions with finite endpoint limits and that compactly supported ReLU splines are dense in 𝐶 0 ⁡ ( ℝ ) . Consequently, each selected element of a single chaotic orbit can be approximated uniformly by a finite ReLU network. Reproducible computations illustrate spline closure, finite orbit block encoding, and endpoint steering. These experiments verify the finite-stage mechanisms used in the proofs without being presented as numerical evidence for an infinite-dimensional topological property.