and the period of the orbit. Fixed points If for some in (that is, the period of the orbit of is ), then is called a fixed point of the iterated sequence. The set of fixed points is often denoted as . There exist a number of fixed-point theorems that guarantee the existence of fixed points in various situations, including the Banach fixed point theorem and the Brouwer fixed point theorem. There are several techniques for convergence acceleration of the sequences produced by fixed point iteration. For example, the Aitken method applied to an iterated fixed point is known as Steffensen's method, and produces quadratic convergence. Limiting behaviour Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an attractive fixed point. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an unstable fixed point. When the points of the orbit converge to one or more limits, the set of accumulation points of the orbit is known as the limit set or the ω-limit set. The ideas of attraction and repulsion generalize similarly; one may categorize iterates into stable sets and unstable sets, according to the behaviour of small neighborhoods under iteration. (Also see Infinite compositions of analytic functions.) Other limiting behaviours are possible; for example, wandering points are points that move away, and never come back even close to where they started. Invariant measure If one considers the evolution of a density distribution, rather than that of individual point dynamics, then the limiting behavior is given by the invariant measure. It can be visualized as the behavior of a point-cloud or dust-cloud under repeated iteration. The invariant measure is an eigenstate of the Ruelle-Frobenius-Perron operator or transfer operator, corresponding to an eigenvalue of 1. Smaller eigenvalues correspond to unstable, decaying states. In general, because repeated iteration corresponds to a shift, the transfer operator, and its adjoint, the Koopman operator can both be interpreted as shift operators action on a shift space. The theory of subshifts of finite type provides general insight into many iterated functions, especially those leading to chaos. Fractional iterates and flows, and negative iterates The notion must be used with care when the equation has multiple solutions, which is normally the case, as in Babbage's