Hub-and-spoke systems from symbolic dynamics can have completely positive mean dimension without uniformly positive mean dimension or entropy, with proofs linking entropy and mean dimension properties at the level of fixed covers.
Ann.\@ Math.\@ (2) 182(1) (2015): 351--389
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Under the stated ranges for k, d and p, the maximal operator given by the sup over r of the absolute value of r^k times the k-th derivative of the spherical mean is bounded on L^p with a bound independent of dimension d.
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Uniformly Positive Mean Dimension
Hub-and-spoke systems from symbolic dynamics can have completely positive mean dimension without uniformly positive mean dimension or entropy, with proofs linking entropy and mean dimension properties at the level of fixed covers.
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Maximal inequalities for derivatives of spherical means
Under the stated ranges for k, d and p, the maximal operator given by the sup over r of the absolute value of r^k times the k-th derivative of the spherical mean is bounded on L^p with a bound independent of dimension d.