Mod[23.234, 1]

f[x_] := Mod[x + a, 1]

a = 0.3

f[1.3]

a = 0.4

f[1.3]

x[0] = 0

Do[x[n + 1] = f[x[n]], {n, 0, 20}]

\(\(Table[x[n], {n, 1, 20}]
\)\)

t1 = Table[{Cos[2*Pi*x[n]], Sin[2*Pi*x[n]]}, {n, 0, 4}]

ListPlot[t1]

a = 0.4123875984365

Do[x[n + 1] = f[x[n]], {n, 0, 200}]

t2 = Table[{Cos[2*Pi*x[n]], Sin[2*Pi*x[n]]}, {n, 0, 200}];

ListPlot[t2]

Do[x[n + 1] = f[x[n]], {n, 0, 2000}]

t2 = Table[{Cos[2*Pi*x[n]], Sin[2*Pi*x[n]]}, {n, 0, 2000}];

ListPlot[t2]

Rationalize[a, 10^(-5)]

\(\(a - 273/662
\)\)

x[662]

x[662 + 662]

x[662 + 661]

x[662 + 663]