Definitions and Examples of Piecewise Affine Dynamics

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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.

1. Is it a good idea to exclude infinite-dimensional examples? (Each cell in a simplicial complex is a subset of some $\mathbb R^k$).
2. Should the triangulation be required to be piecewise linear (automatic in dimension at most 4)?
3. 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 $\mathbb R^\ell$.

# Fundamental Examples

page revision: 9, last edited: 13 Feb 2010 12:00