# Difference Equations & Possible Limits

The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here.

If we look at the behaviour of a point in R n under matrix multiplication, we know that there are only a few pictures (it can remain fixed at a point, go in a circle, go in a spiral, etc.). In projective space, there are even fewer (spirals become circles, etc).

I'm wondering if there are any similar theorems about what pictures are possible in the 'piecewise constant' as opposed to 'constant' case. Precisely, I have an (open) region of R n sliced up into smaller regions by hyperplanes; on each of these regions, I have a (single) matrix. Furthermore, on the boundaries, these matrices agree. I can now run the same sort of dynamics, and I'm curious as to what can happen. In addition to the question as to which 'pictures' are possible, I'm interested in any other general theorems that show up around here - this problem showed up in something else I was doing, and it is a little far from what I usually do.

Thanks!

PS: If the 'piecewise constant' condition seems ridiculous, we can think of the transformation on the line which takes x to 0.7x if x<=0, and takes x to 0.4x if x >=0. These transformations 'look different' at 0 as written, but obviously do the same thing. The piecewise constant condition is essentially the same thing, along more interesting subspaces.

[edited to respond to comment] Thanks for asking, I hope this is more clear: Space is sliced up into regions. Within each region there is a single (constant) matrix. Different regions may have different matrices; however, they must agree on the boundaries.

-
I'm not sure I follow. Are the matrices non-constant within a region? –  Qiaochu Yuan Dec 1 '09 at 16:20
From the example given at the end, I'm guessing the questioner means to ask about "piecewise linear" functions. (Their derivatives are piecewise constant, which I'm guessing is the source of the confusion.) Since this is a broad class of functions, the dynamics can be extremely complicated... If you're interested in specific theorems the Google phrase would be "piecewise linear dynamical systems". –  Martin M. W. Dec 1 '09 at 16:35
add comment

## 1 Answer

Even in the simplest one dimensional case with only "two pieces" you can find all kinds of behavior. Consider the tent functions $T_\alpha\colon[0,1]\to[0,1]$ defined by $$T_\alpha(x)=\alpha(1-|2x-1|),\quad0<\alpha\le1.$$ As $\alpha$ varies from 0 to 1, the dynamics of $T_\alpha$ go from a unique globally atracting fixed point to chaos, just as the logistic family $4\alpha x(1-x)$.

-
Thank you for mentioning this example - I was considering putting it in the original post. On the one hand, this is a clear example that 'bad things happen', since there is no limiting shape to the points traced out. On the other hand, I had some hope that there were theorems about what sort of chaos is possible, and how to rule it out. As another (even simpler) example, if we consider T(x) = x + a mod (1), which is 'linear' on the circle, we get either periodic behaviour or 'chaos', but nothing else, and the chaos is well understood in terms of continued fraction expansions of a. –  First Timer Dec 1 '09 at 21:18
(From looking up the keywords mentioned above, it seems that this type of understanding of chaos is also not available - I just thought it worth mentioning) –  First Timer Dec 1 '09 at 21:19
add comment