  This paper intends to provide new, simple, and self-contained proofs of the
equivalence of various different descriptions of the uniformly hyperbolic
$\mathrm{SL}(2,\mathbb R)$ sequences. While in the scenario of the
Schr\"odinger cocyles, they may in turn be applied to give new and simple
proofs of theorems regarding their relation with the spectral analysis of
one-dimensional discrete Schr\"odinger operators. Concretely, this paper gives
four different descriptions of uniformly hyperbolic sequences together with the
detailed proofs of their equivalence. It provides concise and self-contained
proof of the Johnson's Theorem [Jo], both for sequence and dynamically defined
potentials. In particular, we give two different proofs of the direction
`uniform hyperbolicity away from the spectrum', of which one is the standard
argument in the spirit of Russell Johnson's original proof and the other
involves Combes-Thomas type of estimate [CT]. Finally, the relation between the
Avalanche Principle, discovered by Goldstein-Schlag [GS], and the uniformly
hyperbolic sequence is explored which yields a simple proof and a better
version of the Avalanche Principle. Many ingredients of the proofs in this
paper are new, in particular the use of asymptotic stable and unstable
directions, are of independent interest and have been applied to many other
related problems.
