F[{x_,y_}]={y,((y+1/y)/x)}

SeedRandom[1];
ListPlot[NestList[F,{Random[], Random[]}, 100000]] // Rasterize

SeedRandom[1];
ListPlot[NestList[F,{Random[], Random[]}, 1000000],
PlotRange -> {1.0058,3.4002}+0.0002{{-1,1},{-1,1}}] // Rasterize

SeedRandom[1];
ListPlot[(10000(#-{3.5,3.571-0.266/1000}))& /@ NestList[F,{Random[], Random[]}, 1000000],
PlotRange -> {{-1,1},{-1,1}}] // Rasterize

SeedRandom[1]; p0={Random[], Random[]};
ListPlot[NestList[F,p0, 1000000],
PlotRange -> p0+0.00001{{-1,1},{-1,1}}] // Rasterize