CHAPTER 4, Diffeomorphisms

4.1 Diffeomorphisms.

We now examine certain diffeomorphisms of the real line. The first example is order preserving, whereas the second is order reversing.

f[x_]:=.9Sin[x]+x

Plot[{x,f[x]},{x,0,20},AspectRatio->Automatic,Ticks->None]

f[x_]:=.9Sin[-x]-x+20;
Plot[{x,f[x]},{x,0,20}, AspectRatio->Automatic,Ticks->None]

If we consider (Subscript[f, a])(x)=a sin(x)+x, we can ask the question: for what values of a is (Subscript[f, a]) a diffeomorphism?

Clear[f]
f[x_,a_]:=a*Sin[x]+x
Manipulate[Plot[{x,f[x,a]},{x,0,20}, AspectRatio->Automatic,Ticks->None], {a,-2,2}]

D[f[x,a],x]

This is non-zero for -1<a<1, and (Subscript[f, a]) is a diffeomorphism for these values of a.

Exercises 4.1.

1. For each of the following functions, use a ManipulatePlot to decide whether or not they are homeomorphisms/diffeomorphisms, for different values of the parameter  a  :
(a)  f(x)=(x^3)-a x,     (b)   f(x)=x-a sin (x),   (c)  f(x)= sin (x)-a x.

2.  (a) Verify for different values of n , that the function  f(x)=1+x+(x^2)/2!+(x^3)/3!+...+(x^(2n+1))/(2n+1)!  is strictly increasing with a single root and unique fixed point (so that  f(x) is a diffeomorphism).
(b) in a similar way,verify that   g(x)=1+x+(x^2)/2!+(x^3)/3!+...+(x^(2n))/(2n)!  is concave up and always positive (no zeros).

4.2 Linear Fractional Transformations.

Linear fractional transformations are functions of the form f(x)={a x+b}/{c x+d}, where a d-b c≠0. If d=-a we get an involution, i. e., (f^2)(x)=x for all x in the domain. In the case that d=a, we get a map that is symmetric about the line y=-x.

f[x_]:=(2x+5)/(3x-2)

Plot[{x,f[x]},{x,-6,6},AspectRatio->Automatic,Ticks->None]

f[x_]:=(a*x+b)/(c*x-a)

f[f[x]]

Simplify[f[f[x]]]

f[x_]:=(-x+5)/(3x-1)

Plot[{-x,f[x]},{x,-6,6},AspectRatio->Automatic,Ticks->None]

f[x_]:=(a*x+b)/(c*x+a)

f[-y]

Solve[-x==f[-y],y]

h[a_,x_]:=x^3-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,-3,3}],{a,-4,4}]

Manipulate[Plot[{x,g[a,x]}, {x,-3,3}],{a,-4,4}]

Manipulate[Plot[{x,k[a,x]}, {x,-2,2}],{a,-4,4}]

(g[a,x]-x)/(h[a,x]-x)

Simplify[%]

Factor[%]

r[a_,x_]:=(k[a,x]-x)/(h[a,x]-x)

Simplify[r[x,a]]

Factor[r[x,a]]

Manipulate[Plot[r[a,x], {x,-2,2}],{a,-4,4}]