  There is much latitude between the requirements of Schnirelman's theorem
regarding the ergodicity of individual high-energy eigenstates of classically
chaotic systems on the one hand, and the extreme requirements of random matrix
theory on the other. It seems likely that some eigenstate statistics and
long-time transport behavior bear nonrandom imprints of the underlying
classical dynamics while simultaneously obeying Schnirelman's theorem. Indeed
this was shown earlier in the case of systems which approach classical
ergodicity slowly, and is also realized in the scarring of eigenstates, even in
the $\hbar \to 0$ limit, along unstable periodic orbits and their manifolds.
Here we demonstrate the nonrandom character of eigenstates of Sinai-like
systems. We show that mixing between channels in Sinai systems is dramatically
deficient compared to random matrix theory predictions. The deficit {\it
increases} as $|\log \hbar|$ for $\hbar\to 0$, and is due to the vicinity of
the measure zero set of orbits which never collide with the Sinai obstruction.
Coarse graining to macroscopic scales recovers the Schnirelman result. Three
systems are investigated here: a Sinai-type billiard, a quantum map which
possesses the essential properties of the Sinai billiard, and a unitary map
corresponding to a quasirandom Hamiltonian. Various wavefunction and long-time
transport statistics are defined, theoretically investigated, and compared to
numerical data.
