Invariant Measures for the Logistic Map
Mathematica Hints

(1) In defining the function f(x) = 4 mu x (1-x), you'll want to use the 
delayed set := (so that you can later change the value of mu). We recommend
using f[x_] := ... rather than f[x_,mu_] := ... so as to make Nest and NestList
easier to use.

(2) You should use Nest to iterate onto the attractor (so Nest[f,0.1,100]
will, for example, find f[f[f[...f[0.1]]]]...]), and then use NestList
to generate a series of points on the attractor.

(3) Input <<Graphics`Graphics` to get histograms, which are used like
      hist1 = Histogram[NestList[f,Nest[f...]...], HistogramScale->1]
By giving the graphs names, you can plot experiment versus theory using Show
      theory1 = Plot[...,{x,0,1}]
      Show[hist1, theory1, PlotRange->{0,7}]

(4) You can generate a nice zigzag plot of the first few iterates of the
maximum by using Table: create a table of pairs {f^n(x),n/A} for some 
convenient constant A and use 
      boundaries = ListPlot[...,PlotStyle->RGBColor[0,1,0],PlotJoined->True]
      Show[hist09, boundaries]
to investigate how well the iterates of the maximum track the boundaries.

(5) Generating the bifurcation diagram is a bit tricky. "NestList" does a
nice job of generating iterates, but we want pairs {mu,f^n[mu]}. We
can generate these by taking the transpose of a vector of mu's
      Transpose[{Table[mu, {i, numPoints+1}], NestList[f,Nest[f,xI,...]...]]
You'll then need to do this for several mu, and use Flatten[ListOfLists,1]
to turn the list of lists of points into a list of points. My way of doing
it gave me the following definition:
      dataPoints[numPoints_, numTransient_, xI_, muIterator_] := 
       Flatten[Table[
         Transpose[{Table[mu, {i, numPoints + 1}], 
            NestList[f, Nest[f, xI, numTransient], numPoints]}], muIterator], 1]
Notice that the last argument is an iterator: you need to use the same variable
name "mu" that you used in the definition of f:
      dataPoints[3, 1024, 0.1, {mu, 0., 1., 0.1}]
Using ListPlot with Prolog->AbsolutePointSize[1] will make a plot with smaller
points: doing 64 points per mu and {mu,0.85,1,0.0001} will show most of the
interesting features.

(6) You can use the function dataPoints starting at xI=0.5 and numTransient=1 
to generate the first five boundaries or so: use PlotStyle->RGBColor[1,0,0]
to make them colored, and use Show to superimpose the boundaries onto the
bifurcation diagram.