logistic map averaged values graphed

Question

For the logistic map: https://en.wikipedia.org/wiki/Logistic_map

For a given x, taking the average of n values on the logistic map, gives a converging value.

ie for the x domain 3.56 <= x < 4, with n=100000 averaged values for each x, with a x step size of 0.00001 the graph is:

The Mathematica code:

(*code is slow*) 
Clear["Global`*"]

f[x_] := 4 a*x (1 - x);

xAverages = {};
For[i = 0.89, i < 1, i = i + 0.00001,
 a = i;
 b = Mean[NestList[f, 0.5, 100000]];
 AppendTo[xAverages, b];
 ]

xAverages;
ListPlot[xAverages]

I'm curious about the properties of this graph, ie some x values will converge faster, like the flat sections, which are not chaotic. Also the steep slope sections that form the edges of the flat sections are a transition from the chaotic regions to the non chaotic regions. Can this averaged graph be used to derive the Feigenbaum constant in a similar way to the logistic map as well?

Here is the graph of the logistic map over the same x domain 3.56 <= x < 4 :

ListPlot[Flatten[Table[x = Table[0, {1000}];
   x[[1]] = .01;
   Do[x[[i + 1]] = r*x[[i]] (1 - x[[i]]), {i, 999}];
   {r, #} & /@ x[[-100 ;;]], {r, L1}], 1]]


Flatten[Table[x = Table[0, {1000}];
  x[[1]] = .01;
  Do[x[[i + 1]] = r*x[[i]] (1 - x[[i]]), {i, 999}];
  {r, #} & /@ x[[-100 ;;]], {r, L1}], 1]