Attracting and repelling fixed points and periodic cycles

We call a fixed point $x^{*}$ of the map $f$ attracting if for all $x_{0}$ taken from some neighborhood of the point $x^{*}$, the sequence $x_{n}=f^{n}\left(x_{0}\right)$ converges to the point $x^{*}$. If for some neighborhood $U$ of the fixed point $x^{*}$ and an arbitrary point $x_{0} \in U, x_{0} \neq x^{*}$, there is an index $n$ such that $x_{n} \notin U$, then the point $x^{*}$ is called repelling.

Let $x^{*}$ be a fixed point of the map $f$, and suppose $f$ is differentiable at $x^{*}$. Prove that

For which values of the parameter $a$ and for which $x_{0} \in \mathbb {R}$ does the sequence of iterates $x_{n}=f^{n}\left(x_{0}\right)$ converge, if

The set $\left\{ x_{n}=f^{n}\left(x_{0}\right) \mid n \in \mathbb {N}\right\}$ is called the trajectory of the point $x_{0}$. If the trajectory is finite, then there exists $n \in \mathbb {N}$ such that the points $x_{0}, x_{1}, \ldots , x_{n-1}$ are pairwise distinct, while $x_{n}=x_{0}, x_{n+1}=x_{1}, \ldots$; in this case $x_{0}$ is called a periodic point, the number $n$ its period, and the tuple $\left(x_{0}, \ldots , x_{n-1}\right)$ an $n$-cycle.

Let $\left(x_{0}^{*}, x_{1}^{*}, \ldots , x_{n-1}^{*}\right)-n$-cycle of the map $f$ be given. Prove that

The set $\left\{ x_{n}=f^{n}\left(x_{0}\right) \mid n \in \mathbb {N}\right\}$ is called the trajectory of the point $x_{0}$. If the trajectory is finite, then there exists $n \in \mathbb {N}$ such that the points $x_{0}, x_{1}, \ldots , x_{n-1}$ are pairwise distinct, while $x_{n}=x_{0}, x_{n+1}=x_{1}, \ldots$; in this case $x_{0}$ is called a periodic point, the number $n$ its period, and the tuple $\left(x_{0}, \ldots , x_{n-1}\right)$ an $n$-cycle.

Prove that for $n \geqslant 2$ the Chebyshev polynomial $T_{n}$ (see problems I.2.20 and 1.16 b)), defined on $\left[-1,1\right]$, has cycles of arbitrary order.

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 $f^{n}$.

Find the values of the parameter $a$ for which the map $f(x)=1-a\left|x\right|(x \in \mathbb {R})$ has

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 $f^{n}$.

Let $a>0, f(x)=1-a x^{2}(x \in \mathbb {R})$. Determine for which values of the parameter $a$ an attracting fixed point and an attracting 2-cycle exist. Find the "birth moments" of the 2-, 4- and 3-cycles (i.e., the lower bounds on the values of $a$ for which these cycles exist).

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 $f^{n}$.

Let $f(x)=b x(1-x)(x \in [0,1]), 0<b \leqslant 4$. Find the lower bound of the values of the parameter $b$ for which $f$ has

The set $\left\{ x_{n}=f^{n}\left(x_{0}\right) \mid n \in \mathbb {N}\right\}$ is called the trajectory of the point $x_{0}$. If the trajectory is finite, then there exists $n \in \mathbb {N}$ such that the points $x_{0}, x_{1}, \ldots , x_{n-1}$ are pairwise distinct, while $x_{n}=x_{0}, x_{n+1}=x_{1}, \ldots$; in this case $x_{0}$ is called a periodic point, the number $n$ its period, and the tuple $\left(x_{0}, \ldots , x_{n-1}\right)$ an $n$-cycle.

Let $x_{0} \in (0,1)$ be a fixed point of a continuous map $f:\left[0,1\right] \rightarrow [0,1]$. Prove that if the one-sided derivatives $f_{ \pm }^{\prime }\left(x_{0}\right)$ are negative and $f_{+}^{\prime }\left(x_{0}\right) f_{-}^{\prime }\left(x_{0}\right)>1$, then $f$ has a 2-cycle. In particular, a 2-cycle exists if $f^{\prime }\left(x_{0}\right)<-1$.

The set $\left\{ x_{n}=f^{n}\left(x_{0}\right) \mid n \in \mathbb {N}\right\}$ is called the trajectory of the point $x_{0}$. If the trajectory is finite, then there exists $n \in \mathbb {N}$ such that the points $x_{0}, x_{1}, \ldots , x_{n-1}$ are pairwise distinct, while $x_{n}=x_{0}, x_{n+1}=x_{1}, \ldots$; in this case $x_{0}$ is called a periodic point, the number $n$ its period, and the tuple $\left(x_{0}, \ldots , x_{n-1}\right)$ an $n$-cycle.

Let $g$ be a piecewise smooth map of the interval $\left[0,1\right]$ into itself. Prove that if $\left|g^{\prime }(x)\right| \geqslant a>1$ at all points where the derivative exists, then $g$ has a $2^{n}$-cycle for every $n \in \mathbb {N}$. Show by an example that this fails if the smoothness of the map is violated on a countable set of points.