  It is proved that when R is a local ring of positive characteristic, $\phi$
is its Frobenius endomorphism, and some non-zero finite R-module has finite
flat dimension or finite injective dimension for the R-module structure induced
through $\phi$, then R is regular. This broad generalization of Kunz's
characterization of regularity in positive characteristic is deduced from a
theorem concerning a local ring R with residue field of k of arbitrary
characteristic: If $\phi$ is a contracting endomorphism of R, then the Betti
numbers and the Bass numbers over $\phi$ of any non-zero finitely generated
R-module grow at the same rate, on an exponential scale, as the Betti numbers
of k over R.
