Density function associated with a nonlinear bifurcating map

In the class of nonlinear one-parameter real maps, we study those with bifurcation that exhibits a period doubling cascade. The fixed points of such maps form a finite discrete real set of dimension 2ⁿm, where m is the (odd) number of 'primary branches' of the map in the non-chaotic region and n is a non-negative integer. A new special representation of these maps is constructed that should give more insight into the physical interpretation and enhance their applications in mathematical physics and nonlinear dynamics. We associate with the 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 band structure. The number of density bands is equal to 2^n-1 m unless n = 0 in which case the density has m bands. The location of the bands is independent of the initial state. It depends only on the map parameter and whether the ordering of the fixed points in the set is odd or even. Polynomials orthogonal with respect to this density (weight) function are constructed. The logistic map is taken as an illustrative example.