defines This linear operator is called the transfer operator or the Ruelle–Frobenius–Perron operator. The largest eigenvalue is the Frobenius–Perron eigenvalue, and in this case, it is 1. The associated eigenvector is the invariant measure: in this case, it is the Bernoulli measure. Again, when Spectrum To obtain the spectrum of , one must provide a suitable set of basis functions for the space One such choice is to restrict to the set of all polynomials. In this case, the operator has a discrete spectrum, and the eigenfunctions are (curiously) the Bernoulli polynomials! (This coincidence of naming was presumably not known to Bernoulli.) Indeed, one can easily verify that where the are the Bernoulli polynomials. This follows because the Bernoulli polynomials obey the identity Note that Another basis is provided by the Haar basis, and the functions spanning the space are the Haar wavelets. In this case, one finds a continuous spectrum, consisting of the unit disk on the complex plane. Given in the unit disk, so that , the functions obey for This is a complete basis, in that every integer can be written in the form The Bernoulli polynomials are recovered by setting and A complete basis can be given in other ways, as well; they may be written in terms of the Hurwitz zeta function. Another complete basis is provided by the Takagi function. This is a fractal, differentiable-nowhere function. The eigenfunctions are explicitly of the form where is the triangle wave. One has, again, All of these different bases can be expressed as linear combinations of one-another. In this sense, they are equivalent. The fractal eigenfunctions show an explicit symmetry under the fractal groupoid of the modular group; this is developed in greater detail in the article on the Takagi function (the blancmange curve). Perhaps not a surprise; the Cantor set has exactly the same set of symmetries (as do the continued fractions.) This then leads elegantly into the theory of elliptic equations and modular forms. Relation to the Ising model The Hamiltonian of the zero-field one-dimensional Ising model of spins with periodic boundary conditions can be written as Letting be a suitably chosen normalization constant and be the inverse temperature for the system, the partition function for this model is given by We can implement the renormalization group by integrating out every other spin. In so doing, one finds that can also be equated with the