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Given a polynomial, (or rational function, transcendental entire (meromorphic) function) $f$, and a smooth closed Jordan curve $\gamma$, can we give a complete description of the image of $f(\gamma)$ and $f^{-1}(\gamma)$? It seems very easy at first glance to me. However, I can say almost nothing about the following question: Is there any difference between the case of polynomial maps (or rational maps) and transcendental maps?

I only know that the image or preimage can be represented by sums of piecewise smooth curves, which can not intersect infinitely many times. Can we say something more about it?

What can we say when the curve $\gamma$ is replaced by a Jordan domain $E$ with smooth boundary in the complex plane? The boundary of $f(E)$, the boundary of a different component of $f^{-1}(E)$, or the boundary of unbounded component of the complement of $f^{-1}E$, may be a little complicated.

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You are asking too many questions, some of them are very difficult. Here are some answers.

  1. Image of a Jordan curve under a rational function. Take a circle for $\gamma$. Every continuous function on the circle can be uniformly approximated by a finite trigonometric sum (=Laurent polynomial). Laurent polynomial is a rational function. So the image can be "any" closed curve you want (must be algebraic, of course, but can uniformly approximate any closed curve). If $\gamma$ is not a circle, consider a conformal map $\phi$ of $\gamma$ onto a circle, approximate this conformal map by a polynomial, and then take the composition of your rational function with this polynomial.

  2. For polynomials/entire functions this is not so. Because we have argument principle: The index with respect to any point in the complement of the curve must be zero for a point far away, and non-negative for every other point. So for example, the image cannot have the shape of figure 8. However, a complete topological characterization of images under polynomials is a difficult topological problem. See for example MR0402776 (the review itself contains a nice little survey of the topic). The author credits Titus with examples showing that the index condition is not sufficient. Various combinatorial criteria exist, but they are very complicated.

  3. The difference between polynomials and entire functions you cannot tell (there is no difference in the topology of the image) because polynomials are dense in the space of entire functions.

  4. Then you ask about pre-images. Notice that a pre-image can be disconnected. Components of preimage may be non-Jordan: they are ramified at the critical points of your function.

  5. Components of pre-image of a Jordan region under an entire functions are simply connected (this follows from the maximum princile). Components of the preimage under an entire function can be unbounded.

  6. One can give a complete topological description of pre-images under rational functions and under polynomials, if this is what you want.

EDIT. For example, consider a preimage of a Jordan curve under a polynomial. Take any finite forest (union of disjoint trees) embedded in the plane. Replace every vertex by a topological disk, and let the disks touch wherever there is an edge. This is a topological model of the preimage. If you want to fix the degree of the polynomial, then additionally you prescribe to each disk a positive integer (the degree of the ramified covering), and sum of these integers is the degree of your polynomial. The proof that every such picture can occur is by the Uniformization theorem. (Uniformization theorem reduces all topological questions about preimages to pure topology).

For rational functions the description is similar but more complicated.

For entire functions it is still more complicated because now we not only deal with an infinite forest, but the "discs" can be unbounded, and quite complicated (a disk can have uncountable set of accessible points at infinity, etc.) Here there is an additional difficulty that the Uniformization theorem is not sufficient; one also needs a criterion for distinguishing a disk from the plane.

EDIT 2. However, I strongly believe that in the case of transcendental entire functions, every topological picture consistent with the Maximum Modulus Principle and the evident local condition is possible. I don't think there is a published statement of this but I believe this is not difficult, just a good exercise.

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Thanks for your detailed answer, professor, this is been of great of help for a young student. –  yaoxiao Apr 28 at 14:47
    
Dear professor, can recommend a book or a paper, for which discuss the topic, I have much more interest in it recently. –  yaoxiao Apr 28 at 14:55
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As I repeatedly said you are asking TOO MANY different questions at the same time. If you are interested in IMAGES under polynomials/rational functions, read the papers I mentioned. I do not know a book on the subject. –  Alexandre Eremenko Apr 28 at 15:00

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