Maps with negative Schwarzian derivative

We consider maps $f$ of an interval $\Delta =\left[a, b\right]$ into itself such that $f \in C^{3}(\Delta )$ and the Schwarzian derivative $S f$ (see Problems VII.2.25, VII.2.26 and the definition preceding them) is negative at those points $x$ that are not critical points of $f$ (i.e., where $f^{\prime }(x) \neq 0$). We denote the class of such maps by the symbol $S(\Delta )$. If the points $x_{0}, x_{1}, \ldots , x_{n-1}$, where $x_{k}=f^{k}\left(x_{0}\right)$, are pairwise distinct while $x_{n}=x_{0}$, then the tuple $\left(x_{0}, \ldots , x_{n-1}\right)$ is called an $n$-cycle.

Let $f \in S(\Delta )$ and let the set of critical points of $f$ be finite. Prove that for every $n \in \mathbb {N}$ the transformation $f$ can have only finitely many $n$-cycles.

We consider maps $f$ of an interval $\Delta =\left[a, b\right]$ into itself such that $f \in C^{3}(\Delta )$ and the Schwarzian derivative $S f$ (see Problems VII.2.25, VII.2.26 and the definition preceding them) is negative at those points $x$ that are not critical points of $f$ (i.e., where $f^{\prime }(x) \neq 0$). We denote the class of such maps by the symbol $S(\Delta )$.

Let $f \in S(\Delta ), g=f^{n}(n \geqslant 1), x<y<z$ be consecutive fixed points of the map $g$. Prove that if the interval $\left[x, y\right]$ contains no critical points of $g$, then $g^{\prime }(y)>1$.

We consider maps $f$ of an interval $\Delta =\left[a, b\right]$ into itself such that $f \in C^{3}(\Delta )$ and the Schwarzian derivative $S f$ (see Problems VII.2.25, VII.2.26 and the definition preceding them) is negative at those points $x$ that are not critical points of $f$ (i.e., where $f^{\prime }(x) \neq 0$). We denote the class of such maps by the symbol $S(\Delta )$. A fixed point $x^{*}$ of a map $f$ is called attracting if for all $x_{0}$ from some neighborhood of $x^{*}$ the sequence $x_{n}=f^{n}\left(x_{0}\right)$ converges to $x^{*}$. If the points $x_{0}, x_{1}, \ldots , x_{n-1}$, where $x_{k}=f^{k}\left(x_{0}\right)$, are pairwise distinct while $x_{n}=x_{0}$, then the tuple $\left(x_{0}, \ldots , x_{n-1}\right)$ is called an $n$-cycle; the $n$-cycle is called attracting if $x_{0}$ is an attracting fixed point of $f^{n}$. The set $\left\{ x_{n}=f^{n}\left(x_{0}\right) \mid n \in \mathbb {N}\right\}$ is called the trajectory of the point $x_{0}$.

Let the map $f \in S(\Delta )$ have a unique critical point $c \in (a, b)$. Prove that every attracting cycle of the map $f$ attracts the trajectory of one of the points $a, c, b$, and consequently $f$ can have at most three attracting cycles.

A fixed point $x^{*}$ of a map $f$ is called attracting if for all $x_{0}$ from some neighborhood of $x^{*}$ the sequence $x_{n}=f^{n}\left(x_{0}\right)$ converges to $x^{*}$, and repelling if for some neighborhood $U$ of it and every point $x_{0} \in U$, $x_{0} \neq x^{*}$, there is an index $n$ such that $x_{n} \notin U$. If the points $x_{0}, x_{1}, \ldots , x_{n-1}$, where $x_{k}=f^{k}\left(x_{0}\right)$, are pairwise distinct while $x_{n}=x_{0}$, then the tuple $\left(x_{0}, \ldots , x_{n-1}\right)$ is called an $n$-cycle and $n$ the period of $x_{0}$; the $n$-cycle is called attracting if $x_{0}$ is an attracting fixed point of $f^{n}$, and repelling if $x_{0}$ is a repelling fixed point of $f^{n}$.

Prove that a map of the form $f(x)=1-a x^{2}(x \in \mathbb {R})$, for $a \leqslant 2$, cannot have more than one attracting cycle, while for $a=2$ it has an $n$-cycle for every $n \in \mathbb {N}$, but none of them is attracting.