On the Use of the Schwarzian derivative in Real One-Dimensional Dynamics

The Schwarzian derivative was first formulated in 1869 by Hermann A. Schwarz in his work on conformal mappings. A first use in one-dimensional dynamics was made by David Singer in 1978 applying it to the study of $C^{3}$ maps from the unit interval to itself [4]. A first approach was made by D.J. Allwright in [1] in the study of bifurcations for $C^{3}$ maps that satisfy a certain property denoted by $P$, which resembles the Schwarzian derivative. Singer showed that the assumption of a negative Schwarzian derivative implies the existence of a finite number of attracting periodic orbits, each of which attracts some critical point or some boundary point, and each neutral periodic point is attracting. In later progress, it was found that maps with negative Schwarzian derivative also possess local properties that are useful for establishing certain distortion bounds, particularly when focusing on cross ratios [3] [5] [2].

At first sight, negative Schwarzian derivative assumption may appear somewhat arbitrary since it does not seem to be a dynamical condition. In this work, we show that this assumption is not arbitrary in any sense, but rather strictly related to a sufficient condition that guarantees the fulfillment of the Minimum Principle for the derivative of the map and its iterates, which is the key point in the proof of Singer’s Theorem. To the best of our knowledge, this is a simple illustrative explanation about the use of the Schwarzian derivative in the context of one- dimensional differential dynamics.

An important result in one-dimensional differential dynamics was stated by Singer in [4]. This result shows that $C^{3}$ maps in the interval, satisfying certain differentiable condition involving the derivatives up to the third order, have a finite number of attracting periodic points, and that each neutral point is indeed an attractor.

Before stating we recall some necessary definitions. We say that the $p$ is a periodic point for a map $f$ if for some integer $n$, $f^{n}(p)=p.$ The basin of a periodic point $p$ is the set of points whose $\omega-$limit set contains $p$. A periodic point $p$ is attracting and that $O(p)$ is an attracting periodic orbit if its basin contains an interval that contains $p$. The immediate basin of a periodic point $p$ is the union of the connected components of its basin which contain a point from $O(p)$. The periodic point $p$ is called a hyperbolic attractor if $|(f^{n})^{\prime}(p)|<1$, a hyperbolic repeller if $|(f^{n})^{\prime}(p)|>1$, super attractor if $|(f^{n})^{\prime}(p)|=0$ and neutral if $|(f^{n})^{\prime}(p)|=1$. Finally, we say that $c$ is a critical point of $f$ if $f^{\prime}(c)=0$. A critical point is called non-degenerate if $f^{\prime\prime}(c)\neq 0$.

It follows from the definition that any map $g$ satisfying the Minimum Principle cannot have a negative local maximum or a positive local minimum. If the Minimum Principle holds for the derivative $f^{\prime}$ of a differentiable map $f$, then significant dynamical implications occurs, particularly regarding the type and number of positively oriented fixed points that map $f$ can have, indeed, the immediate basin of any positively oriented attracting fixed point contains either a critical point of $f$ or a boundary point, and each neutral positively oriented fixed point is actually attracting.

On the other hand, if $f$ is a differentiable map such that $f^{\prime}$ satisfies the Minimum Principle, then a non-degenerate critical point for $f^{\prime}$ is either a positive local maxima or a negative local minima for $f^{\prime}$, in other words, for $f^{\prime}$ satisfying the Minimum Principle, if $f^{\prime\prime}(x)=0$ and $f^{\prime\prime\prime}(x)\neq 0$ then ${\displaystyle\frac{f^{\prime\prime\prime}(x)}{f^{\prime}(x)}}$ is negative. Conversely, if any non-vanish critical point for $f^{\prime}$ is non-degenerate (i.e. if $f^{\prime\prime}(x)=0$ implies $f^{\prime\prime\prime}(x)\neq 0$) and ${\displaystyle\frac{f^{\prime\prime\prime}(x)}{f^{\prime}(x)}}$ is negative, then the derivative $f^{\prime}$ satisfies the Minimum Principle.

It is important to note that the key point in the proof of Singer’s theorem is that for all positive integers $n$, $(f^{n})^{\prime}$ satisfies the Minimum Principle, which implies that for the derivative $(f^{n})^{\prime}$ local maxima are positive and local minima are negative (see Proposition 2.4 in [4]).

Keeping this in mind, we aim to find a condition on $f$ such that, for all positive integers $n$, the derivative $(f^{n})^{\prime}$ satisfies the Minimum Principle. Unfortunately, establishing a condition with these characteristics does not seem to be a straightforward computation. Even more, the dynamical implications of the Minimal Principle for negatively oriented fixed points are not entirely clear to us, unlike what happens in the case of positively oriented fixed points, so we begin by considering derivative of $f^{2}$.

Let consider a differentiable map $f$ defined on an interval $I$, such that any non-vanish critical point for $(f^{2})^{\prime}$ is non-degenerate, in other words, for any point $x\in I$ such that $(f^{2})^{\prime}(x)\neq 0$ and $(f^{2})^{\prime\prime}(x)=0$ then $(f^{2})^{\prime\prime\prime}(x)\neq 0$. By the Chain rule, we have the following

Consequently,

and

Thus, from (2.1) and (2)

Now, since we have assumed $(f^{2})^{\prime\prime}(x)=0$, from (2.2), then we obtain

First, from (2.5) we have the following

and replacing by the second term of the right member of (2.4) gives the following

Second, again from (2.5) we have the following

and replacing in the second term of the right member of (2.4) gives the following

Thus, adding (2.6) and (2.7) we obtain

Finally, dividing by 2 and put together terms

Hence, if we assume that both expressions

and

are negative then, by(2), the expression ${\displaystyle\frac{(f^{2})^{\prime\prime\prime}(x)}{(f^{2})^{\prime}(x)}}$ is negative whenever $(f^{2})^{\prime\prime}(x)=0$. So, the Minimum Principle is guaranteed for $(f^{2})^{\prime}$.

More generally, consider a differentiable map $f$ defined on an interval $I$, such that given a positive integer $n$, any non-vanish critical point $x\in I$ of $(f^{n+1})^{\prime}$ is non-degenerate, in other words, given a positive integer $n$, any point $x\in I$ such that $(f^{n+1})^{\prime}(x)\neq 0$, and $(f^{n+1})^{\prime\prime}(x)=0$ then $(f^{n+1})^{\prime\prime\prime}(x)\neq 0$. Following a similar computation as before, we obtain the following

Since, we have assumed that $(f^{n+1})^{\prime\prime}(x)=0$, we have

First, from (2.12) we have the following

and replacing by the second term of the right member of (2.11) gives the following

Second, again from (2.12) we have the following

and replacing in the second term of the right member of (2.11) gives the following

Thus, adding (2.13) and (2.14), dividing by 2 and put together terms we obtain

Hence, if we assume that both expressions

and

are negative then, by (2), the expression ${\displaystyle\frac{(f^{n+1})^{\prime\prime\prime}(x)}{(f^{n+1})^{\prime}(x)}}$ is negative whenever $(f^{n+1})^{\prime\prime}(x)=0$. So, again the Minimum Principle is guaranteed for the derivative $(f^{n})^{\prime}$ for any positive integer $n$.

The expression that appears in (2.9) and (2.16) is known as the Schwarzian derivative of $f$ in $x$, and is denoted by $Sf(x)$. In the same way, the expression in (2.10) and (2.17) corresponds to $Sf(f(x))$ and $Sf^{n}(f(x))$ respectively.

In the context of one-dimensional dynamics, the Schwarzian derivative was introduced by David Singer in 1978, applying it to the study of $C^{3}$ maps from the unit interval to itself. An initial approach was made by D.J. Allwright in [1] in the study of bifurcations for $C^{3}$ maps. Singer was the first to observe that the formula for the Schwarzian derivative for the composition of two functions is given by

which follows to straightforward computation [4], indeed expressions (2) and (2) clearly suggest this. In particular, if a map has a negative Schwarzian derivative, so do all its iterates. Thus, if we assume that the Schwarzian derivative of $f$ is negative everywhere, then the expression in (2.17) is also negative. This makes it a valuable tool in the study of one-dimensional dynamics.

In conclusion, the assumption of a negative Schwarzian derivative for a map is far from arbitrary, it is directly associated with the fulfillment of the Minimum Principle for the derivative of the map as well as for its iterates, and its dynamical implications, as we have explained.

To the best of our knowledge, in the literature, there is no a satisfactory explanation about the use of the Schwarzian derivative in Real One-Dimensional Dynamics. Therefore, we believe that this note is valuable in order to shed light on this point.