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# Generalities

First, some ludicrously general definitions.

*Abbreviation.* **Piecewise affine** will be abbreviated to *PA*.

A **general PA dynamics** is defined by a group (or semigroup or groupoid or semi-groupoid) acting by PA transformations.

A **general PA transformation** $f$ is a partially defined self-map of a PA manifold $M$, i.e., a topological set with a given triangulation $h:K\to M$. A point is *regular* for such a transformation if the map is defined on a neighborhood $U$ of it such that (i) $U$ and $f(U)$ are each contained in a cell of the triangulation; (ii) $h^{-1}\circ f\circ h| h^{-1}(U)$ is an affine map of some open subset of some $\mathbb R^k$. Non regular points are called **singular** and define $\operatorname{sing}(f)$ the singular set.

*Questions.*

- Is it a good idea to exclude infinite-dimensional examples? (Each cell in a simplicial complex is a subset of some $\mathbb R^k$).
- Should the triangulation be required to be piecewise linear (automatic in dimension at most 4)?
- Many results (and proofs) can be generalized to piecewise
*projective*dynamics. Can one define an interesting class of piecewise ''simple'' transformations?

*Remark.* The singular set (which includes points at which the map is not defined) should be small in some sense to justify the PA terminology.

A **regular PA transformation** is a general PA transformation such that (i) the simplicial complex underlying the triangulation is a homogeneous simplicial $k$-complex, for some $k\in\mathbb N$, (ii) the singular set is a union of simplices of dimension strictly less than $k$.

A **simple PA transformation** is a regular PA transformation with a finite triangulation.

An **everywhere PA transformation** (or *EPA transformation*) is a regular PA transformation which is defined everywhere with the property that, for the interior $U$ of any cell, $h^{-1}\circ f\circ h| h^{-1}(U)$ is an affine map of some open subset of some $\mathbb R^\ell$.

# Basic Classes of Interest

## Low dimension

## Uniform hyperbolicity

## Zero Entropy

- Piecewise isometries (including dual billiards)