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The Bernoulli polynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there.
==Bernoulli map==
The [[Bernoulli map]] is a certain simple model of a [[dissipative]] [[dynamical system]], describing the effect of a [[shift operator]] on an infinite string of coin-flips (the [[Cantor set]]). The Bernoulli map is a one-sided version of the closely related [[Baker's map]]. The Bernoulli map generalizes to a [[p-adic|k-adic]] version, which acts on infinite strings of ''k'' symbols: this is the [[Bernoulli scheme]]. The [[transfer operator]] <math>\mathcal{L}_k</math> corresponding to the shift operator on the Bernoulli scheme is given by
:<math>[\mathcal{L}_k f](x) = \frac{1}{k}\sum_{n=0}^{k-1}f\left(\frac{x+n}{k}\right)</math>
Perhaps not surprisingly, the eigenvectors of this operator are given by the Bernoulli polynomials. That is, one has that
:<math>\mathcal{L}_k B_m = \frac{1}{k^m}B_m</math>
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.
==Characteristic zero==
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