CHAPTER 5, Other Cantor Sets.

5.1 Cantor Sets Arising From the Logistic Maps.

We now look at Cantor sets arising from the quadratic maps. These sets arise from (Subscript[L, r])(x)=r x(1-x), 0≤x≤ 1,  where r is a parameter, 4≤ r≤ 5.

L[x_,a_]:=a*x(1-x)

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

Manipulate[Plot[{1,g[x,a]},{x,0,1},AspectRatio->Automatic],{a,4,5}]

As we increase the parameter through values greater than 4, we see that those points that stay mapped inside [0,1] resemble the first few stages in the construction of a Cantor set.  Mathematica has a built in construction of the Cantor Staircase function (see Exercise 5.3 # 2). This is an increasing function, constant on each component of the complement of the Cantor set

Plot[CantorStaircase[x],{x,0,1}]

5.2 Cantor Sets Arising From the Tent Maps.

In a similar way, we can look at Cantor sets arising from tent maps. We have seen the graph of the tent map:

t[x_]:=Piecewise[{{x, x<1/2}, {(1-x), 1/2<x}}]

Plot[t[x],{x,0,1}]

If we execute the ManipulatePlot command on each of the following, again we see that the set of points that stay mapped inside the unit interval form the first stages in the construction of a Cantor type set.

p[x_,a_]:=a*t[x]

r[x_,a_]:=p[p[x,a],a]

Manipulate[Plot[{1,p[x,a]},{x,0,1},AspectRatio->Automatic],{a,0,4}]

Exercise 5.2

For the function s(x) defined below, graph s and use ManipulatePlot as in the previous example to investigate some of its dynamical properties (fixed points, period-2 points etc.).

s[x_]:=Piecewise[{{x^2, x<1/2}, {(1-x)^2, 1/2<x}}]

Plot[s[x],{x,0,1}]

u[x_,a_]:=a*s[x]

v[x_,a_]:=u[u[x,a],a]

w[x_,a_]:=u[u[u[x,a],a],a]

Manipulate[Plot[{x,1,u[x,a]},{x,0,1},AspectRatio->Automatic],{a,0,4}]

Manipulate[Plot[{x,1,v[x,a]},{x,0,1},AspectRatio->Automatic],{a,0,5}]

Manipulate[Plot[{x,1,w[x,a]},{x,0,1},AspectRatio->Automatic],{a,0,5}]