© Hindawi Publishing Corp. FAÀ DI BRUNO’S FORMULA AND NONHYPERBOLIC FIXED POINTS OF ONE-DIMENSIONAL MAPS

Fixed-point theory of one-dimensional maps of ℝ does 
not completely address the issue of nonhyperbolic fixed points. 
This note generalizes the existing tests to completely classify 
all such fixed points. To do this, a family of operators are 
exhibited that are analogous to generalizations of the Schwarzian 
derivative. In addition, a family of functions f are exhibited 
such that the Maclaurin series of f ( f ( x ) ) and x are identical.


Introduction.
The study of dynamics of maps from R to R is central to many fields including discrete dynamical systems [1, 3, 5, 12], difference equations [4, 7, 9, 10], and differential equations via Poincaré [11] and Lorenz maps [1, 5, 8].It is well known that a fixed point of such a map can be of three types.A stable fixed point attracts nearby points towards it, under iteration.An unstable fixed point repels nearby points, whereas a semistable fixed point attracts nearby points on one side (say, to the left), and repels nearby points on the other side.
Until recently, the classification of fixed points has been incomplete.Specifically, no test existed if f (x * ) = x * , f (x * ) = 1, f (x * ) = 0, and f (x * ) = 0. Also, no test existed if f (x * ) = x * , f (x * ) = −1, Sf (x * ) = 0, where Sf (x) is the Schwarzian derivative, defined in this case as Sf (x) = −f (x * )−1.5(f (x * )) 2 .This situation was recently remedied in [2], which demonstrated a sequence of tests that fill both gaps.However, the tests for the second gap have certain inadequacies.This note restates and improves on the solution for the second gap.This is summarized in Figure 1.1.
The gray area to the lower left is from [2], and the gray boxed area marked "NEW" is from this note.

Summary of previously known results.
Let f : R → R be a continuous map, and x * such that f (x * ) = x * .We say that x * is semistable from the left (resp., right) if, given > 0, there is δ > 0 such that x * − δ < x < x * (resp., for any positive number of iterations of f .If x * is semistable from both sides, we say that x * is stable, whereas if it is semistable from neither side, we say that x * is unstable.In the sequel, when we say a fixed point is semistable, we imply that it is not stable.Theorem 2.1 [2].
Classification of fixed points of a one-dimensional map.
(1) If k is even and A > 0, then x * is semistable from the left.
(2) If k is even and A < 0, then x * is semistable from the right.
(3) If k is odd and A > 0, then x * is unstable.
(4) If k is odd and A < 0, then x * is stable.
This classification of nonoscillatory nonhyperbolic fixed points (i.e., where f (x * ) = 1) was used in [2] to generate a test for oscillatory nonhyperbolic fixed points (where f (x * ) = −1), which satisfy Sf (x * ) = 0 and therefore were previously unclassified.This test is due to the following classical theorem.
. Then x * is classified under f in the same way as under g.
Observe that for oscillatory nonhyperbolic fixed points, g( This allows us to classify g (and, hence, f ) using Theorem 2.1.Furthermore, [2] contains the following result, proved using Taylor's theorem.
Theorem 2.3 [2].Let f : R → R be continuous with f (x * ) = x * , and f (x * ) = −1.Then x * is either stable or unstable; it cannot be semistable.This method can be improved, as we will see in the sequel.
3. Another method.One of the drawbacks of the previous algorithm for the case of a fixed point with f (x * ) = −1 is the need to pass to g(x).To study an nth degree polynomial with coefficients bounded by N, we need to consider an n 2 -d degree polynomial with coefficients bounded by 24, and zero from then on.We first find Sf (0) = −f (0) − 1.5(f (0)) 2 = 0. Unfortunately, this falls in the gap of the classical theory.Therefore, to classify 0, the previous algorithm requires us to pass to the substantially more complicated g 9 , to find that g (5) (0) = (64)(5!)= 7680 > 0, making 0 an unstable fixed point.
An improvement to the algorithm is made possible by a formula published in the mid-nineteenth century by Faà di Bruno.For a history of this result as well as some biographical information, see [6].
for f ∈ C n and where a = a 1 +a 2 +•••+a n and the sum extends over all possible integer In our context, we are evaluating it all at the fixed point x * , with g(x) = f (f (x)) and f (x * ) = −1.The sum then becomes The character of this result will be more evident with several examples.

Generalized Schwarzian-type derivatives.
Using formula (3.2), we can calculate generalized analogues of the Schwarzian derivative to use in our classification.We take S k f (x) = (1/2)g (2k+1) (x), and simplify using the assumption that S i f (x) = 0 for all i < k: (10) This allows a simpler algorithm to classify oscillatory nonhyperbolic fixed points of a one-dimensional map.(1) if A > 0, then x * is unstable, (2) if A < 0, then x * is stable.
Theorem 4.1 allows classification of fixed points with simpler calculation.Additional Schwarzian-type derivatives are simple to calculate using formula (3.2).However, in general this will not be necessary, as each generalized derivative is only needed if all earlier ones are zero.

Theorem 4.4. S k f (x) has exactly one term containing an odd derivative of f , and that term is −f (2k+1) (x).
Proof.First, recall that S k f (x) = (1/2)g (2k+1) (x).We now use formula (3.2) with n = 2k + 1. Observe that the highest derivative that can appear is f (n) (x * ).This can appear in a term in only two ways: if a = n or if a n > 0. The restrictions on the sum force exactly two terms containing f (n) Now, we prove by strong induction that no other terms appear with odd derivatives of f .The case k = 1 corresponds to the classical Schwarzian derivative.For k > 1, we observe that we are simplifying under the assumption that S i f (x) = 0 for all i < k.The result holds for these S i f (x) by the induction hypothesis.Hence, we can solve for each odd derivative f (2i+1) (x) in terms of even derivatives, and substitute into S k f (x).
We now use these Schwarzian-type derivatives to generate a class of functions, each of which is analytically a "square root" of the identity at the origin.

A special class of functions. The function h(x) =
x has a natural square root, namely, f (x) = −x.By this we mean that f (f (x)) = h(x) = x.However, we can construct an infinite class of other functions f , each of which is analytically a square root of h(x) at the origin.That is, The most general square root is given by the following power series: We observe that f (0) = 0, f (0) = −1, and otherwise f (n) (x) = a n .In order to ensure that (f • f ) (n) (0) = h (n) (0) = 0, we must ensure that Sf k (0) = 0 for each k.We can do this inductively by choosing the odd derivatives (a 2j+1 ) as per Theorem 4.4.The even derivatives (a 2j ) may be chosen freely.For example, we may choose a 2 freely, but then

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Theorem 4 . 1 .
Let f : R → R be continuous with f (x * ) = x * and f (x * ) = −1.Let k ≥ 1 be minimal such that S k f (x * ) = A ≠ 0. Then x * is classified as follows: