The Flat-Topped Tent Map

We consider the flat-topped tent map, parametrized by the value h between 0 to 1.

T[x_]:=Piecewise[{{2*x, 0 ≤ x ≤ 1/2}, {2*(1-x), 1/2 < x ≤ 1}}]

TFlat[h_][x_]:=Min[{T[x], h}]

ListofIterates[h_]:=Flatten[Table[{{Nest[TFlat[h], h, k], Nest[TFlat[h], h, k]}, {Nest[TFlat[h], h, k], Nest[TFlat[h], h,  k+1]}}, {k, 1, 100}], 1]
Manipulate[Plot[{TFlat[h][x], x}, {x,0, 1},PlotStyle->{{Thick, Red}, {Dashed, Thick,  Black}},  Prolog->{Darker[Blue], Thickness[.004], Line[ListofIterates[h]]}, PlotLegends->"Length Orbit" LengthOrbit[h] ], {h, 0, .99999, .001}]

LengthOrbit[h_]:=Length[DeleteDuplicates[NestList[TFlat[h],h,  100]]]

LengthOrbit[.83]

Bifurcation Diagram for the Flat Tent

The parameter h from 0 to 1 is along the x-axis; the y-axis is the unit interval.  We plot the long term behavior of TFlat[h][x] as a bifurcation diagram.  
The main diagonal is the line x = h.

ListPlot[Table[Table[{h, Nest[TFlat[h], h, k]},  {k, 0, 100}], {h, 0, 1, .0005}]]

Reduce[Nest[TFlat[h], h, 2]== h  && 0 ≤ h ≤1, h]

Solve[Nest[TFlat[9/10], x0, 2]== x0  && 0 ≤ x0 ≤1, x0]

Reduce[Nest[TFlat[h], h, 2]== h && Nest[TFlat[h], h, 1]≠  h&&  0 ≤ h ≤1, h]

Reduce[Intersection[{Reduce[Nest[TFlat[h], h, 4]== h&&   0 ≤ h ≤1, h],  Reduce[Nest[TFlat[h], h, 2]≠  h&&   0 ≤ h ≤1, h]}]]

Reduce[Intersection[{Reduce[Nest[TFlat[h], h, 3]== h&&   0 ≤ h ≤1, h],  Reduce[Nest[TFlat[h], h, 1]≠  h&&   0 ≤ h ≤1, h]}]]

Reduce[Intersection[{Reduce[Nest[TFlat[h], h, 5]== h&&   0 ≤ h ≤1, h],  Reduce[Nest[TFlat[h], h, 1]≠  h&&   0 ≤ h ≤1, h]}]]

Reduce[Intersection[{Reduce[Nest[TFlat[h], h, 6]== h&&   0 ≤ h ≤1, h],  Reduce[Nest[TFlat[h], h, 3]≠  h&&Nest[TFlat[h], h, 2]≠  h &&   0 ≤ h ≤1, h]}]]

Reduce[Intersection[{Reduce[Nest[TFlat[h], h, 7]== h&&   0 ≤ h ≤1, h],  Reduce[Nest[TFlat[h], h, 1]≠  h&&   0 ≤ h ≤1, h]}]]