Multiplication theorem: Difference between revisions

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It is the fact that the eigenvalues <math>k^{-m}<1</math> that marks this as a dissipative system: for a non-dissipative [[measure-preserving dynamical system]], the eigenvalues of the transfer operator lie on the unit circle.

One may construct a function obeying the multiplication theorem from any [[totally multiplicative function]]. Let <math>f(n)</math> be totally multiplicative; that is, <math>f(mn)=f(m)f(n)</math> for any integers ''m'', ''n''. Define its Fourier series as

More generally, this operator also has a continuous spectrum: the Hurwitz zeta function <math>\zeta(s,x)</math> is also an eigenvector, with eigenvalue <math>k^s</math>. Note that the Bernoulli polynomials arise as limiting cases of the Hurwitz zeta for integer values of ''-s''.

:<math>g(x)=\sum_{n=1}^\infty f(n) \exp(2\pi inx)</math>

Assuming that the sum converges, so that ''g''(''x'') exists, one then has that it obeys the multiplication theorem; that is, that

:<math>\frac{1}{k}\sum_{n=0}^{k-1}g\left(\frac{x+n}{k}\right)=f(k)g(x)</math>

That is, ''g''(''x'') is an eigenfunction of Bernoulli transfer operator, with eigenvalue ''f''(''k''). The multiplication theorem for the Bernoulli polynomials then follows as a special case of the multiplicative function <math>f(n)=n^{-s}</math>.

==Characteristic zero==