Piecewise Expanding Maps

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

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