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We discuss spreading estimates for dynamical systems given by the iterationof an extended CMV matrix. Using a connection due toCantero--Gr\"unbaum--Moral--Vel\'azquez, this enables us to study spreadingrates for quantum walks in one spatial dimension. We prove several generalresults which establish quantitative upper and lower bounds on the spreading ofa quantum walk in terms of estimates on a pair of associated matrix cocycles.To demonstrate the power and utility of these methods, we apply them to severalconcrete cases of interest. In the case where the coins are distributedaccording to an element of the Fibonacci subshift, we are able to rathercompletely describe the dynamics in a particular asymptotic regime. As apleasant consequence, this supplies the first concrete example of a quantumwalk with anomalous transport, to the best of our knowldege. We also proveballistic transport for a quantum walk whose coins are periodicallydistributed.

Spreading Estimates for Quantum Walks on the Integer Lattice via Power-Law Bounds on Transfer Matrices