Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function, find the function such that Schröder's equation is an eigenvalue equation for the composition operator, which sends a function to. If is a fixed point of, meaning, then either or. Thus, provided that is finite and does not vanish or diverge, the eigenvalue is given by.
Functional significance
For, if is analytic on the unit disk, fixes, and, then Gabriel Koenigs showed in 1884 that there is an analytic satisfying Schröder's equation. This is one of the first steps in a long line of theorems fruitful for understanding composition operators on analytic function spaces, cf. Koenigs function. Equations such as Schröder's are suitable to encoding self-similarity, and have thus been extensively utilized in studies of nonlinear dynamics. It is also used in studies of turbulence, as well as the renormalization group. An equivalent transpose form of Schröder's equation for the inverse of Schröder's conjugacy function is. The change of variables further converts Schröder's equation to the older Abel equation,. Similarly, the change of variables converts Schröder's equation to Böttcher's equation,. Moreover, for the velocity,, Julia's equation, , holds. The -th power of a solution of Schröder's equation provides a solution of Schröder's equation with eigenvalue, instead. In the same vein, for an invertible solution of Schröder's equation, the function is also a solution, for anyperiodic function with period. All solutions of Schröder's equation are related in this manner.
Solutions
Schröder's equation was solved analytically if is an attracting fixed point, that is by Gabriel Koenigs. In the case of a superattracting fixed point,, Schröder's equation is unwieldy, and had best be transformed to Böttcher's equation. There are a good number of particular solutions dating back to Schröder's original 1870 paper. The series expansion around a fixed point and the relevant convergence properties of the solution for the resulting orbit and its analyticity properties are cogently summarized by Szekeres. Several of the solutions are furnished in terms of asymptotic series, cf. Carleman matrix.
Applications
It is used to analyse discrete dynamical systems by finding a new coordinate system in which the system generated by h looks simpler, a mere dilation. More specifically, a system for which a discrete unittime step amounts to, can have its smooth orbit reconstructed from the solution of the above Schröder's equation, its conjugacy equation. That is,. In general, all of its functional iterates are provided by the orbit for real — not necessarily positive or integer. The set of, i.e., of all positive integer iterates of is called the splinter of. However, all iterates of are likewise specified through the coordinate transformationdetermined to solve Schröder's equation: a holographic continuous interpolation of the initial discrete recursion has been constructed; in effect, the entire orbit. For instance, the functional square root is, so that, and so on. For example, special cases of the logistic map such as the chaotic case were already worked out by Schröder in his original article, In fact, this solution is seen to result as motion dictated by a sequence of switchback potentials,, a generic feature of continuous iterates effected by Schröder's equation. A nonchaotic case he also illustrated with his method,, yields Likewise, for the Beverton–Holt model,, one readily finds , so that
