The Gingerbreadman map

Clear[x,y,n]
iter=40000;

sol=RecurrenceTable[{
x[n+1]==1-y[n]+Abs[x[n]],
y[n+1]==x[n],
x[0]==-0.3,y[0]==0},{x,y},{n,0,iter}];
fixedpoints=Solve[{x==1-y+Abs[x],y==x},{x,y}];
Print[Style["Fixed point coordinates",19,Blue]]
coordinates={x,y}/.fixedpoints
fixedp=Graphics[{PointSize[0.01],Red,Point[coordinates]}];
Rasterize[Show[ListPlot[sol,PlotMarkers->{"•",3},Frame-> True,FrameLabel->{x,y},LabelStyle->20,PlotRange->{{-3,8},{-3,8}},AspectRatio->0.9,ImageSize->600],fixedp]]

ListLinePlot[sol[[All,1]],FrameLabel->{n,"(Subscript[x, n])"},PlotStyle->Thin,Frame -> True,PlotRange->{{0,0.1 iter},{-5,8}},LabelStyle->20,ImageSize->600,AspectRatio->0.3]
ListLinePlot[sol[[All,2]],FrameLabel->{n,"(Subscript[y, n])"},PlotStyle->Thin,Frame -> True,PlotRange->{{0,0.1 iter},{-5,8}},LabelStyle->20,ImageSize->600,AspectRatio->0.3]

Creation of this notebook was supported by IT Academy program of Information Technology Foundation for Education (HITSA).