page_title: Iterated function
section_title: Conjugacy
hierarchical_section_title: Iterated function / Conjugacy
context_page_description: In mathematics, an iterated function is a function X → X which is obtained by composing another function f: X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial number, the result of applying a given function is fed again in the function as input, and this process is repeated.
  Iterated functions are objects of study in computer science, fractals, dynamical systems, mathematics and renormalization group physics.
context_section_description: If f and g are two iterated functions, and there exists a homeomorphism h such that   g =  h⁻¹ ○ f ○ h ,  then f and g are said to be topologically conjugate.
  Clearly, topological conjugacy is preserved under iteration, as gⁿ = h⁻¹  ○ f ⁿ ○ h.  Thus, if one can solve for one iterated function system, one also  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, one has 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: functional iteration has been reduced to multiplication!  Here, however, the exponent n no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit: the monoid of the Picard sequence (cf. transformation semigroup) has generalized to a full continuous group.
  This method (perturbative determination of the principal eigenfunction Ψ, cf. Carleman matrix) is equivalent to the algorithm of the preceding section, albeit, in practice,  more powerful and systematic.
caption: Iterates of the sine function (blue), in the first half-period. Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it; and further fractional iterates up to the 1/64th. The functions below the (blue)  sine are six integral iterates below it, starting with the second iterate (red) and ending with the 64th iterate. The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin).     (From the general pedagogy web-site.[18] For the notation, see [2].)