CHAPTER 21, Interval Exchange Transformations.

21.1 Interval Exchange Transformations

An interval exchange transformation takes an interval such as [0,1]), cuts it up into finitely many subintervals and maps it piecewise linearly onto itself. For example, here is such a map involving two intervals. It is actually the irrational rotation of the interval [0,1), defined by (Subscript[T, α ])(x)=x+α mod 1, where α=1-π/5.

f[x_]:= Piecewise[{{x+1-Pi/5, x<Pi/5}, {(x-Pi/5), Pi/5<x}}]

Plot[f[x],{x,0,1},AspectRatio->Automatic]

Plot[f[f[x]],{x,0,1},AspectRatio->Automatic]

Note that the iterates are also interval exchanges of two intervals. Again this is true for the third iterate:

Plot[f[f[f[x]]],{x,0,1},AspectRatio->Automatic]

A ManipulatePlot on a general two-interval exchange will give us the opportunity to see what the second and third iterates look like.

h[a_,x_]:=Piecewise[{{x+1-a, x<a}, {(x-a), a<x}}]

g[a_,x_]:=h[a,h[a,x]]

k[a_,x_]:=h[a,h[a,h[a,x]]]

Manipulate[Plot[{x,h[a,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{a,0,1}]

We can follow iterations using the webPlot module. Below we iterate x=1/2, ten times.

webPlot[g_,{x_,xmin_,xmax_},{a1_,n_}, opts_]:=Module[{seq, r, pts, web, graph}, r[t_]:=N[g/.x->t];seq=NestList[r,a1,n]; pts=Flatten[Table[{{seq[[i]],seq[[i+1]]},{seq[[i+1]],seq[[i+1]]}}, {i,1,n}],1]; web=Graphics[{Hue[0], Line[PrependTo[pts, {seq[[1]], 0}]]}];graph=Plot[{x,r[x]}, {x, xmin, xmax}, DisplayFunction->Identity]; Show[web, graph, opts, Frame->True, PlotRange->{{xmin, xmax}, {xmin, xmax}}]];

webPlot[f[x],{x,0,1},{1/2,10},AspectRatio->Automatic]

21.2 Exchanges of Three Intervals.

f[a_,b_,x_]:=Piecewise[{{x+1-a, x<a}, {(x-a+1-b), a<x<b},{x-b,b<x<1}}]

We have a two-parameter function giving rise to an interval exchange of three intervals (in general), where we require a<b to have an interval exchange transformation.

Manipulate[Plot[{f[a,b,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{a,0,1},{b,0,1}]

m[a_,b_,x_]:=f[a,b,f[a,b,x]]

Manipulate[Plot[{m[a,b,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{a,0,1},{b,0,1}]

n[a_,b_,x_]:=f[a,b,f[a,b,f[a,b,x]]]

Manipulate[Plot[{n[a,b,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{a,0,1},{b,0,1}]

h[a_,x_]:=Piecewise[{{-x+1, x<a}, {(x-a), a<x}}]

g[a_,x_]:=h[a,h[a,x]]

k[a_,x_]:=h[a,h[a,h[a,x]]]

Manipulate[Plot[{x,h[a,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{a,0,1}]

21.3 Interval Exchange Transformations with a Flip.

These are similar to intervals exchange transformations, but the orientation is reversed on some of the intervals. Here is an example using two intervals.

l[x_]:=Piecewise[{{-x+1, x<1/Sqrt[2]-.3}, {(x-1/Sqrt[2]+.3), 1/Sqrt[2]-.3<x}}]

Plot[l[x],{x,0,1},AspectRatio->Automatic]

Plot[l[l[x]],{x,0,1}, AspectRatio->Automatic]

webPlot[l[x],{x,0,1},{2/3,5},AspectRatio->Automatic]

We see there are 4-cycles, and in fact (f^4)(x)= identity, on a two large intervals (see below). Also we see that (f^12) is the identity map.

Plot[l[l[l[x]]],{x,0,1}, AspectRatio->Automatic]

Plot[l[l[l[l[x]]]],{x,0,1}, AspectRatio->Automatic]

Plot[l[l[l[l[l[x]]]]],{x,0,1}, AspectRatio->Automatic]

g[x_]:=l[l[l[l[l[l[x]]]]]]

Plot[g[x],{x,0,1}, AspectRatio->Automatic]

Plot[g[g[x]],{x,0,1}, AspectRatio->Automatic]

Here is a ManipulatePlot of a transformation involving two intervals with a flip:

l[a_,x_]:=Piecewise[{{-x+1, x<a}, {(x-a), a<x}}]

g[r_,x_]:=l[r,l[r,x]]

k[r_,x_]:=l[r,l[r,l[r,x]]]

Manipulate[Plot[{x,k[r,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{r,0,1}]

Here is a ManipulatePlot for a transformation involving three intervals with a flip (again we need a<b):

t[a_,b_,x_]:=Piecewise[{{x+1-a, x<a}, {(1-x), a<x<b},{x-b,b<x<1}}]

Manipulate[Plot[{t[a,b,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{a,0,1},{b,0,1}]

Exercises 21.1

1. Prove that the iterates of an interval exchange of two intervals, is an exchange of two intervals (Use Mathematica to look at specific examples). Is this true for exchanges of n>2 intervals (i.e. if we iterate an exchange of n>2 intervals, do we get an exchange of (Subscript[n,  ]) intervals) ? (Hint: Look at some of the examples above).

2. Prove that if the graph of an interval exchange transformation is symmetric about the line y=1-x, then it is conjugate to its inverse (actually this is true for any such symmetric map). Deduce that interval exchange maps of two and three intervals are conjugate to their inverses. Note that if the graph of f(x) is symmetric with respect to y=1-x, then so is the graph of all of its iterates. Note that our example of a flip map on two intervals is conjugate to its inverse.

3. Use the ManipulatePlot for the two-interval flip map to show that there is a 2-cycle created when a=1/3, and a 3-cycle when a=1/4. Also note that for a>1/3, (f^12)=identity. More generally, is it true that a flip map f that is conjugate to its inverse has (f^p)=identity, for some p>1?
 
4. Investigate the iterations of three interval exchanges with flips.