set and expanded around the fixed point value of 2 is then an infinite series, which, taking just the first three terms, is correct to the first decimal place when n is positive–cf. Tetration: . (Using the other fixed point causes the series to diverge.) For , the series computes the inverse function . Example 3 With the function , expand around the fixed point 1 to get the series which is simply the Taylor series of x(bn ) expanded around 1. Conjugacy If and are two iterated functions, and there exists a homeomorphism such that , then and are said to be topologically conjugate. Clearly, topological conjugacy is preserved under iteration, as . 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 , one has the iteration of as , for any function . Making the substitution yields , a form known as the Abel equation. Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at = 0, (0) = 0, one may often solve Schröder's equation for a function Ψ, which makes locally conjugate to a mere dilation, , that is . Thus, its iteration orbit, or flow, under suitable provisions (e.g., ), amounts to the conjugate of the orbit of the monomial, , where in this expression serves as a plain exponent: functional iteration has been reduced to multiplication! Here, however, the exponent 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. Markov chains If the function is linear and can be described by a stochastic matrix, that is, a matrix whose rows or columns sum to one, then the iterated system is known as a Markov chain. Examples There are many chaotic maps. Well-known iterated functions include the Mandelbrot set and iterated function systems. Ernst Schröder, in 1870, worked out special cases of the logistic map, such as the chaotic case , so that , hence . A nonchaotic