U’(x) = U(phi(x)) suffers from aliasing in the signal processing sense; it is a staircase function with jumps whenever x mod 10 = 0.
While the original paper isn’t super explicit about this, phi is a stochastic map so the only way to define U’ is as an average over the distribution phi(x); therefore, it need not jump like this (phi can smoothly change the probability distribution, leading to smooth changes in U’).
While the original paper isn’t super explicit about this, phi is a stochastic map so the only way to define U’ is as an average over the distribution phi(x); therefore, it need not jump like this (phi can smoothly change the probability distribution, leading to smooth changes in U’).