A piecewise expanding map is a simple PA whose restriction to the interior of each facet of the triangulation uniformly expands some distance. More precisely, there is a distance $latex d$ on the affine manifold $latex M$ and a constant $latex \lambda>1$ such that, for any $latex x,y$ in the interior of some facet, $latex d(fx,fy)\geq d(x,y)$.
In this segment we do not consider extra assumptions, except continuity or genericity.
The theory is most complete (though see questions below).
Both maximal entropy measures and physical measures can be studied by a unified and powerful thermodynamical formalism. These results relate ergodic properties with the set of periodic orbits through Ruelle zeta functions.
Using on Markov extensions
Using functional determinants
See small boundary condition