It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps. 


I think it is interesting to ask whether we can prove it for polynomial case by using elementary method.



Is there any elementary proof of No wandering domain for polynomials
If by "elementary method", you mean a proof that avoids using techniques from quasiconformal methods, then the short answer is "no". Giving a different proof of the No Wandering Domains (NWD) theorem, which works without dynamical assumptions (e.g., hyperbolicity, non-recurrence etc) is a well-known open problem. This is true even for quadratic polynomials.


Let me subvert your question a little bit, and ask whether there is a **different** (not necessarily **more elementary**) proof. The answer is still "no" in general, but becomes more interesting.


Sullivan's proof is one of those mysterious arguments that we encounter sometimes which prove a desired result, but do not really tell us in a satisfying way why it is true. When I first learnt this proof as a postgraduate student, from Misha Lyubich in Stony Brook, I was totally dumbfounded by the idea of using the finite-dimensionality of parameter space. Misha told me that, when the proof was first announced, his reaction was much the same. 


There is, nonetheless, another method of proving absence of wandering domains, and this is to prove **rigidity**, the absence of **any** nontrivial deformations of the map. Of course, this is usually much harder than Sullivan's argument, and in general it is still open. Indeed, it would imply the famous **hyperbolicity conjecture**, which is open even for the Mandelbrot set. On the other hand, it has been solved for many classes of systems, including all real polynomials with only real critical values.


Now, these arguments still use the theory of quasiconformal mappings, but they do **not** essentially rely on finite-dimensionality of parameter space. Indeed, in our paper "Absence of wandering domains for some real entire functions with bounded singular sets" with Mihaljević-Brandt (DOI 10.1007/s00208-013-0936-z; <http://arxiv.org/abs/1104.0034>), we use rigidity for real-analytic functions to establish NWD for some real entire functions with infinite singular sets, without any dynamical assumptions. (The assumptions are that the function is real with all singular values real, plus some geometric assumptions on the functions itself, but crucially no assumptions on the iterates of singular values.) 


Another way of saying this is that, when/if **Local Connectivity of the Mandelbrot Set** will be finally established, it is very likely that we will have a new proof of NWD for quadratic polynomials.


The point, in the end, is that we do not understand the behaviour of a completely general rational map or polynomial well enough to prove key facts such as rigidity. If we did, we could likely give a more natural (but more difficult) prove of NWD. Sullivan's argument is a brilliant shortcut that allows us to get away without such a detailed knowledge: while we cannot exclude the presence of *all* quasiconformal deformations at present, having a wandering domain actually would lead to far too many of them (in the case of a finite-dimensional deformation space). 


Of course, this does not rule out the existence of a more elementary argument for NWD - but it seems unlikely (at least without also making a key breakthrough in our general understanding of rational/polynomial dynamics).


