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We present a general framework for weak convergence to decorated L\'evyprocesses in enriched spaces of c\`adl\`ag functions for vector-valuedprocesses arising in deterministic systems. Applications include uniformlyexpanding maps and unbounded observables as well as nonuniformlyexpanding/hyperbolic maps with bounded observables. The latter includesintermittent maps and dispersing billiards with flat cusps. In many of theseexamples, convergence fails in all of the Skorohod topologies. Moreover, theenriched space picks up details of excursions that are not recorded by Skorohodor Whitt topologies.

Convergence to decorated L\'evy processes in non-Skorohod topologies for dynamical systems