  We consider discrete one-dimensional Schr\"odinger operators whose potentials
are generated by H\"older continuous sampling along the orbits of a uniformly
hyperbolic transformation. For any ergodic measure satisfying a suitable
bounded distortion property, we establish a uniform large deviation estimate in
a large energy region provided that the sampling function is locally constant
or has small supremum norm. We also prove full spectral Anderson localization
for the operators in question.
