m = {{1, 1}, {1, 2}}

Mod[m . {1, 2}, 1]

x[0] = {2/10, 3/10};
nmax = 2000;
Do[x[n + 1] = Mod[m . x[n], 1], {n, 0, nmax}];
ListPlot[Table[x[n], {n, 1, nmax}], Frame -> True]

x[0] = {0.2, 0.3}
nmax = 2000;
Do[x[n + 1] = Mod[m . x[n], 1], {n, 0, nmax}];
ListPlot[Table[x[n], {n, 1, nmax}], Frame -> True]

x[0] = {2/3, N[Pi]}; nmax = 2000;
Do[x[n + 1] = Mod[m . x[n], 1], {n, 0, nmax}];
ListPlot[Table[x[n], {n, 1, nmax}], Frame -> True]

f[{x_, y_}] := Mod[m . {x, y}, 1]; f[{0.3, 0.2}]

ListPlot[NestList[f, {0.35, 0.4}, 100]]

Do[
      ListPlot[Flatten[
          Table[Nest[f, {0.01*mm, 0.01*nn}, ww], {mm, 0, 10}, {nn, 0, 10}], 
          1], PlotRange -> {{0, 1}, {0, 1}}, 
        PlotJoined -> True], {ww, 0, 10}]