Let be intervals and let l, r be the components of . Then the Cross Ratio is defined as
Assume g is a monotone function on the interval t, and g(t)=T, g(j)=J, g(l)=L, g(r)=R. Then define
Define the Schwarzian Derivative for function g,
Proposition 1. Assume f and g are functions, then
Proposition 2. If for some and , then for all .
Proposition 3. Minimum Principle.
Assume , is a diffeomorphism with negative schwarzian derivative, then
Theorem 1. Real Koebe Principle.
Let Sf<0. Then for any intervals and any n for which is a diffeomorphism one has the following. If contains a scaled neighbourhood of , then
Moreover, there exists a universal function which does not depend on f, n, and t such that
Theorem 2. Complex Koebe Principle
Suppose that contains a scaled neighbourhood of the disc . Then for any univalent function one has a universal function which only depends on such that
Theorem 3. Schwarz Lemma (Original Form)
Assume is the unit disc on the complex plane , is a holomorphic function with . Then for all and Moreover, if for some or then for some
Assume is the unit disc on the complex plane , and is a holomorphic function, then
Assume is the upper half plane of the complex plane , is a holomorphic map. Then
Corollary 3. Pick Theorem
The hyperbolic metric on is , assume denotes the hyperbolic distance between and on . Assume is a holomorphic function, then
Moreover, if for some points , then , where
Background in hyperbolic geometry
where is an interval. It is easy to show that is conformally equivalent to the upper half plane and define as
k is determined by the external angle at which the discs intersect the real line. Moreover, Define
Corollary 4. (NS) Schwarz Lemma
(1) Assume is a holomorphic map, then
(2) Assume is a real polynomial map, its critical points are on the real line. Assume is a diffeomorphism, then there exists a set such that and
is a conformal map.
Assume is a univalent map and D contains scaled neighbourhood of and assume f maps the real line to the real line. For each there exists such that if J is a real interval in , then
The Hyperbolic Metric On the Real Interval and Cross Ratio
As far as we know, the hyperbolic metric on the unit disc is
Then the restriction to the real line is
Moreover, from it, we can deduce the hyperbolic metric on the real interval is
If , then the hyperbolic length of the interval on the total interval is
Theorem 4. Assume is a diffeomorphism with negative schwarzian derivative. Assume , then
That means f expands the hyperbolic metric on the real interval.
Proof. Since the schwarzian derivative of f is negative,
Therefore, That means f expands the hyperbolic metric on the real interval.
Remark. From Schwarz-Pick Theorem, for a holomorphic map , contracts the hyperbolic distance in the unit disc . Conversely, from above, for a diffeomorphism with negative schwarzian derivative, expands the hyperbolic distance in the real interval.
Exercise 1. “Mathematical Tools for One Dimensional Dynamics” Exercise 6.5, Chapter 6
Let be a diffeomorphism without fixed points ( being a closed interval on the real line). If for all , then there exists a unique such that for all .
Proof. If is a decreasing map, then the right boundary of the real interval I is the . Therefore, assume that is an increasing map on the real interval I.
Since has no fixed points on the real interval I, then or for all . Without lost of generality, assume for all . Since is a continuous function on the closed interval I, there exists such that for all .
By contradiction, there exist two distinct points such that and for all . From here, we know that .
From Langrange’s mean value theorem, there exists such that . Since the schwarzian derivative of is negative, from the minimal principle, we get
i.e. . However, from the definition of and , we get
This is a contradiction. Therefore, the existence of is unique.
Assume is a diffeomorphism, define the non-linearity of as
Theorem 5. Koebe Non-linearity Principle.
Given , there exists such that, if is a diffeomorphism into the reals and for all then we have
for all Show that as (This recovers the classical Koebe non-linearity principle).