On the mixing coefficients of piecewise monotonic maps
  We investigate the mixing coefficients of interval maps satisfying Rychlik's conditions. A mixing Lasota-Yorke map is reverse $\phi$-mixing. If its invariant density is uniformly bounded away from 0, it is $\phi$-mixing iff all images of all orders are big in which case it is $\psi$-mixing. Among $\b$-transformations, non-$\phi$-mixing is generic. In this sense, the asymmetry of $\phi$-mixing is natural. 
Jon Aaronson, Hitoshi Nakada