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.

# Dimension one

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

# Dimension two

## Physical measures

Buzzi, Tsujii

## Entropy theory

See small boundary condition

# Higher dimensions

## Physical measures

Gora-Boyarski, Saussol

Buzzi, Cowieson

Tsujii

## Entropy theory

???