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    Ph.D. Projects.
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As an example of what can occur the reader might like to consider the 
following family

$$f_{a}(x)=1-{a+1\over x}+{a\over x^{2}}.$$
We restrict to real parameter $a$ and real variable $x$, so 
considering $f_{a}$ as a function on the real line $\mathbb R$. 
corresponding family of complex functions wth complex parameter is 
very interesting but this will suffice for  the moment.

The function 
$f_{a}$ has a vertical asymptote at $x=0$, a horizontal asymptote at 
$y=1$, and a critical (or stationary) point at $x={2a\over a+1}$. 
Regarded as a holomorphic map of the Riemann sphere, $0$ is another 
critical point of $f_{a}$, but just considering $f_{a}$ as a 
real-valued function, we simply note that 
$$\lim _{x\to 0}f_{a}(x)=\infty ,$$
which we write as $f_{a}(0)=\infty $, and
$$\lim _{x\to \infty }f_{a}(x)=1$$
which we write as $f_{a}(\infty )=1$. We also have $f_{a}(1)=0$.

It is a fact that, in the theory of complex dynamics, the dynamics is 
often completely determined by the dynamics of the critical pionts. 
Therefore, we are particularly interested in the sequence
$$f_{a}^{n}\left( {2a\over a+1}\right) $$
for different values of $a$, where $f_{a}^{n}$ denotes the $n$-fold 
iterate. There are some special values of $a$ where this sequence is 
particularly simple:
$$a=1:\  {2a\over a+1}=1, {\rm{\ so\ }}f_{1}^{3}\left( {2a\over 
a+1}\right) =f_{1}^{3}(1)=1,$$
$$a=-1:\  {2a\over a+1}=-\infty , {\rm{\ so\ }}f_{-1}^{3}\left( {2a\over 
a+1}\right) =f_{-1}^{3}(\infty )=\infty .$$
Also we note that for all $a$
$$f_{a}\left( {2a\over a+1}\right) =-{(a-1)^{2}/4a}.$$
The equation
$${2a\over a+1}==-{(a-1)^{2}/4a}$$
has just one real root $a_{1}$, which is $<0$. Note that
$$f_{a_{1}}'\left( {2a_{1}\over a_{1}+1}\right) =0.$$

The sequence $f_{a}^{n}( {2a\over a+1}) $ is then also particularly 
simple in somewhat larger sets:
$$a<a_{1}+\delta _{1},\ \ -1-\delta _{1}<a<0,\ \ 0<a<1+\delta _{3},$$
for some numbers $\delta _{j}>0$, $j=1$, $2$, $3$. For 
$-1-\delta _{1}<a<0$, 
$$\lim _{n\to \infty }f_{a}^{3n}({2a\over a+1})=\infty ,$$
for $0<a<1+\delta _{3}$
$$\lim _{n\to \infty }f_{a}^{3n}({2a\over a+1})=1,$$
and for $a<a_{1}+\delta _{1}$, $f_{a}^{n}({2a\over a+1})$ converges 
to an attractive fixed point. 

The sets 
$$a_{1}+\delta _{1}\leq a\leq -1-\delta _{1},\ \ 1+\delta _{3}\leq a$$ 
are more interesting.
Write
$$X_{1}=[-\infty ,0],\ X_{2}=[0,1],\ X_{3}=[1,\infty ].$$
Then for $a\leq -1$,
$$f_{a}(X_{1})=X_{1}\cup X_{2},\ \ f_{a}(X_{2})=X_{1},\ \ 
f_{a}(X_{3})\subset X_{2}\cup X_{3}.$$
For $a\geq 1$,
$$f_{a}(X_{1})=X_{3},\ \ f_{a}(X_{2})=X_{2}\cup X_{3},\ \ 
f_{a}(X_{3})\subset X_{1}\cup X_{2}.$$

One can then ask what are the possible {\em{symbolic dynamics}} of 
the sequence $f_{a}^{n}({2a\over a+1})$, that is, what are the 
possibilities for sequences $X_{i_{n}}$ ($n\geq 0$) where 
$f_{a}^{n}({2a\over a+1})\in X_{i_{n}}$. For $a< -1$, the 
the sequences which occur are fairly easily characterised as 
follows: $X_{i_{j}}=X_{3}$ for $j\leq k$, some $k\geq 0$, and 
$X_{i_{k+1}}=X_{2}$, $X_{i_{j}}=X_{1}$ or $X_{2}$, such that if 
$X_{i_{j}}=X_{2}$ then $X_{i_{j+1}}=X_{1}$. For $a\geq 1$ it is 
similar but rather more tricky: the start of the sequence imposes a 
condition on the rest of the sequence. For $a\leq -1$ the nature of 
the sequence means that it is impossible for ${2a\over a+1}$ to be 
periodic for $a<-1$, although it is possible for ${2a\over a+1}$ to 
be eventually periodic, for example to have $f_{a}^{n}({2a\over 
a+1})=0$ for some $n>0$. For $a>1$ it is possible for ${2a\over a+1}$ to 
be periodic.

These calculations in themselves are just that - nontrivial but a few 
hours' or days' work, not weeks or months on end. Looking at the real 
case in isolation is not the best strategy. Also, the coding used above 
is not the only possible one. There are other codings which recognise 
hidden Julia sets -- more than one -- and the relationship between 
the different codings could be clarified. Lookin at real parameter 
certaunly gives some information, but there is a much better 
description if one moves to the complex parameter, and uses more 
sophisticated methods. These methods have not yet been used on other 
families. So there is a lot of development possible. 

There is also an analytical question which can probably be tackled now. 

{\em{Problem: Is the set of parameter values for which 
$f_{a}^{n}({2a\over a+1})$ converges to a periodic orbit dense in 
$a<-1$ (or in $a>1$)?}}

It is quite likely that the methods used to prove the corresponding 
result for real quadratic polynomials will work here. The real 
quadratic polynomial result was proved by Grazcyk and Swiatek, and by 
Lyubich, in the 1990's, and there have been a number of important 
extensions in the last few years by van Strien and Weixao Shen and 
Lyubich and collaborators, for example. But this would need checking.

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