Question

what are rotational number ?give any three example​

Answer 1

Answer:

DEFINE

Suppose that f: S1 → S1 is an orientation preserving homeomorphism of the circle S1 = R/Z. Then f may be lifted to a homeomorphism F: R → R of the real line, satisfying

{\displaystyle F(x+m)=F(x)+m}

for every real number x and every integer m.

The rotation number of f is defined in terms of the iterates of F:

{\displaystyle \omega (f)=\lim _{n\to \infty }{\frac {F^{n}(x)-x}{n}}.}

Henri Poincaré proved that the limit exists and is independent of the choice of the starting point x. The lift F is unique modulo integers, therefore the rotation number is a well-defined element of R/Z. Intuitively, it measures the average rotation angle along the orbitsof f.

EXAMPLES

If f is a rotation by 2πθ (where 0≤θ<2π), then

{\displaystyle F(x)=x+\theta ,}

then its rotation number is θ (cf Irrational rotation).

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