Iterated Function - Conjugacy

Conjugacy

If f and g are two iterated functions, and there exists a homeomorphism h such that, then f and g are said to be topologically conjugate. Clearly, topological conjugacy is preserved under iteration, as one has that, so that if one can solve one iterated function system, one has solutions for all topologically conjugate systems. For example, the tent map is topologically conjugate to the logistic map. As a special case, taking f(x) = x + 1, we have the iteration of g(x) = h−1(h(x) + 1) as gn(x) = h−1(h(x) + n), for any function h. Making the substitution x = h−1(y) = ϕ(y) yields g(ϕ(y)) = ϕ(y + 1), a form known as the Abel equation.

Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at x = 0, f(0) = 0, one may often solve Schröder's equation for a function Ψ, which makes f(x) locally conjugate to a mere dilation, g(x) = f '(0) x, that is f(x) = Ψ−1(f '(0) Ψ(x)).

Thus, its iteration orbit, or flow, under suitable provisions (e.g., f '(0) ≠ 1), amounts to the conjugate of the orbit of the monomial, Ψ−1(f '(0)n Ψ(x)), where n in this expression serves as a plain exponent. Here, however, n no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit: the semigroup of the Picard sequence has generalized to a full continuous group.

This method (perturbative determination of Ψ, cf Carleman matrix) is equivalent to the algorithm of the preceding section, albeit, in practice, more powerful and systematic.