In the class of nonlinear one-parameter real maps we study those with
bifurcation that exhibits period doubling cascade. The fixed points of such a
map form a finite discrete real set with dimension (2^n)m, where m is the (odd)
number of "primary branches" of the map in the non-chaotic region and n is a
non-negative integer. We associate with this map a nonlinear dynamical system
whose Hamiltonian matrix is real, tridiagonal and symmetric. The density of
states of the system is calculated and shown to have a number of separated
bands equals to (2^n-1)m for n not equal 0, in which case the density has m
bands. The location of the bands depends only on the map parameter and the
odd/even ordering of the fixed points in the set. Polynomials orthogonal with
respect to this density (weight) function are constructed. The logistic map is
taken as an illustrative example.