Dble[x_] = Piecewise[{{2x, x≥0 && x< (1/2)},{2x-1, x≥ (1/2) && x< 1}}]

Table[Nest[Dble[#] &,1/2,k],{k,0,20}]

Table[Nest[Dble[#] &,1/3,k],{k,0,20}]

Table[Nest[Dble[#] &,1/4,k],{k,0,20}]

Table[Nest[Dble[#] &,1/5,k],{k,0,20}]

Table[Nest[Dble[#] &,1/6,k],{k,0,20}]

Table[Nest[Dble[#] &,1/7,k],{k,0,20}]

Table[Nest[Dble[#] &,1/8,k],{k,0,20}]

Table[Nest[Dble[#] &,1/9,k],{k,0,20}]

Table[Nest[Dble[#] &,1/10,k],{k,0,20}]

Table[Nest[Dble[#] &,1/(2^4),k],{k,0,20}]

Table[Nest[Dble[#] &,1/(2^9),k],{k,0,20}]

Table[Nest[Simplify[Dble[#]] &,1/Sqrt[2],k],{k,0,20}]

XPlot = Plot[x,{x,0,1}, PlotStyle->{{Red,Thickness[0.01]}}];

Dbleplot = Plot[Dble[x],{x,0,1}, PlotStyle->{{Blue,Thickness[0.01]}}];

Show[XPlot,Dbleplot]

Dble2 = Simplify[Dble[Dble[x]]]

Dbleplot2 = Plot[Dble2,{x,0,1}, PlotStyle->{{Blue,Thickness[0.01]}}];

Show[XPlot,Dbleplot2]

Solve[x==-1+4 x,x]

Solve[x==-2+4 x,x]

Dble[1/3]

Dble[2/3]

Dble3 = Simplify[Dble[Simplify[Dble[Dble[x]]]]]

Dbleplot3 = Plot[Dble3,{x,0,1}, PlotStyle->{{Blue,Thickness[0.01]}}];

Show[XPlot,Dbleplot3]

Dble4 = Simplify[Dble[Simplify[Dble[Simplify[Dble[Dble[x]]]]]]]

Dbleplot4 = Plot[Dble4,{x,0,1}, PlotStyle->{{Blue,Thickness[0.01]}}];

Show[Dbleplot4, XPlot]

2046/11

256-2-2-12

240/8