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\title{Fractional iterations of functions.}
\begin{document}
\section*{7. Conjugacy and iterations}
Functions $f$ and $g$ are called {\it conjugated} whenever $f=R\circ
g\circ R^{-1}$ for some function $R$.

{\bf Exercise.} Show that in this case $f^{(n)}=R\circ g^{(n)}\circ
R^{-1}$.

\noindent We use notation $f \sim g$ whenever $\frac
{f(x)}{g(x)}\xrightarrow[]{x\to 0}1$.

\begin {enumerate}
\item a) Show that for any $n\in \mathbb Z$ the function $\cos (n\arccos(x))$ is a polynom, and that any two functions of this kind commute one with each
other.\\
b) Show that for any $n\in \mathbb Z$ the function
$\sin((2n+1)\cdot\arcsin(x))$ is a polynom, and that any two
functions
of this kind commute one with each other.\\
c) Show that for any $n\in \mathbb Z$ the function $\tan
(n\arctan(x))$ is {\it rational}, i.e. is a quotient of two
polynoms. Show that any two functions of this kind commute one with
each other.

\noindent {\bf Remark.} Parts a and b provides nontrivial examples
of commuting families of polynoms. By the deep theorem of Reed any
other nontrivial family of commuting polynoms essentially coincides
with a or b.

\item Show that the function $\sin x$ is  not conjugated to a polynom.

\item Find fractional iterations of functions $\frac {ax+b}{cx+d}$ for any $a, b, c, d$.

Hence we can explicitly describe fractional iterations of linear
functions, we wish to connect them with as many fractional
iterations of other functions as possible. Essentially we wish to
find a big enough class of functions $f$ for which exists a {\it
conjugating function} $R$ such that \begin{center}$R\circ f\circ
R^{(-1)}$\end{center} is a linear function. From time to time it is
reasonable to find such conjugating function in some neighborhood of
some point.

\item Let $f$ be a function such that $f(0) = 0, f'(0) = k$. Evaluate $(f^{(n)})'
(0)$. Assume that $f$ is conjugated to $lx$ for some number $l$ with
some smooth (i.e. infinitely differentiable) conjugating function
$R$. Prove that in this case $k=l$. If $|k| < 1$ then $f^{(n)}(x)
\xrightarrow[]{n\to\infty} 0$ in for all $x$ from some neighborhood
of 0. In this case we call 0 an {\it attracting} point of $f$. If
$|k|>1$ we call 0 a {\it repelling} point of $f$.

\item a) Let $0$ be an attracting point of a continuously differentiable function $f$.
Prove that for all $x_0$ from some neighborhood of 0 the limit
$$\lim\limits_{n\to\infty}\frac{f^{(n)}(0)}{k^n}=G(x_0)$$ exists.
Prove that $G$ is continuous and that $G(k\cdot
G^{(-1)}(x))=f(x)$.\\b) Prove that $G$ is continuously
differentiable.\\c*) Prove that if $f$ smooth then $G$ is smooth.

\item Prove the following equality.
     $$
     \frac{\sqrt{x}}{2}\cdot\frac{\sqrt{2+\sqrt{x}}}{2}\cdot
     \frac{\sqrt{2+\sqrt{2+\sqrt{x}}}}{2}\cdots=
     \frac{4-x^2}{\sqrt{2\ln(\frac{x+\sqrt{x^2-4}}{2})}}.
     $$

{\bf Definition.} Let $M\subset\mathbb R$ be a set. We call $x\in M$
a {\it limit point} of $M$ if any neighborhood of $x$ contains
infinitely many points of $M$. We call the set $\{
f^{(n)}(x)\}_{n\in\mathbb Z_{\ge0} }$ an {\it orbit of $x$ under the
action of $f$}.

\item Let $f, g$ be a pair of commuting smooth functions such that f$(0)=g(0)=0$ and $f(x) \sim x^{\lambda}, g(x) \sim x^{\delta}$. Then $f =
g^{(\log_{\delta}\lambda)}$ (Here you have to define fractional
iterations in a spirit of the introduction).

\item We call two points $x_0, x_1$ of an invertible function $f$ {\it
neighbor} if there exists a point $x$ such that
$\lim\limits_{n\to-\infty}f^{(n)}(x)=x_0$ and
$\lim\limits_{n\to+\infty}f^{(n)}(x)=x_1$.

Let $x_0$ and $x_1$ be common neighbor fixed points of commuting
invertible continuously differentiable functions $f$ and $g$. Assume
$x_0$ is attracting and $x_1$ is repelling for both $f$ and $g$.
Prove that in this case
    $$\log_{|g'(x_0)|}|f'(x_0)|=\log_{|g'(x_1)|}|f'(x_1)|.$$

\item Prove that functions of problems 6.1, 6.3, 6.5, 6.7 are fractional iterations of the corresponding functions.

\item Let $f$ be a decreasing function such that $f(0) = 0$ and $f(x) \neq x$ for all $x \neq 0$. Does there exists an infinite family $\mathfrak F$ of pairwise noncommuting functions such that any element of $\mathfrak F$ commutes with $f$?

\end {enumerate}
\section*{8. More about polynoms}
\begin{enumerate}

        \item Let $P(x)$ be a polynom of degree $n>1$. Then for all
        $m$ the set of polynoms of degree $m$ such that they commute
        with $f$ is finite.

\item Let $P(x)$ be a polynom of degree $n> 1$, $Q(x)$ be a polynom of degree $m>
    1$ such that \\a) $P\circ Q=Q\circ P$, b) $P(x_0)=Q(x_0)=x_0$, c)$P'(x_0)>1$, d) in any punctured neighborhood of $x_0$ there exists a point $x_i$
    such that $P^{(k)}(x_i)\xrightarrow[]{k\to\infty}\infty$.\\ Prove
    that $P'(x_0)^{\log_n(m)}=Q'(x_0)$.

\end{enumerate}

{\bf Remark.} We note that the condition "to be commutative" is {\it
algebraic}, i.e. is equivalent to a system of polynomial equations
on coefficients. Therefore we could assume that the value of
derivatives in all fixed point are algebraic. In assumption of
transcendency of powers $\alpha^{\log_n(m)}$, where $\alpha$ is some
algebraic number and $n$ is not a rational power of $\alpha$, we
have that $k$ is a rational power of $n$. It is interesting to
derive the classification of commuting polynoms from this
observation. This problem is really valuable because it provides a
connection between dynamical systems, theory of transcendence
numbers and theory of Diofant approximations.


\end{document}
{\bf Notation.} If $|f(x)| \leq |g(x)|$ for all small enough
positive $x$ we write $f \preceq g$. Note that if $f \not\preceq g$
and $g \not\preceq f$ then $f$ and $g$ are incomparable.
