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There are many simpler sequences which are not orbits, for example, it is impossible to have $a_n=a_{n+1}\neq a_{n+k}$ for some $k>1$.

This can be generalized to $a_n=a_{n+m}\neq a_{n+km},\; k>1$. In fact such sequences are not orbits of ANY function.

Other conditions can be obtained if $a_n$ has a limit, and tends to it with aproximately geometric speed. Then this limit must be an attracting point, and using the local linearization at this point one obtains VERY strong conditions on the sequence.

EDIT. A related question was studied by Fatou. Instead of a rational function he considers a germ $f(z)=\lambda z+\ldots,\; |\lambda|<1$. He takes an orbit $a_n$ and makes a generating function $$g(\zeta)=\sum_{n=0}^\infty a_n\zeta^n.$$ He proves that $g$ is meromorphic in the whole plane with poles at the geometric progression $\lambda^{-k},\; k\geq 0$.

Fatou, P. Sur une classe remarquable de séries de Taylor. Ann. de l’Éc. Norm. (3) 27, 43-53 (1910).

There are many simpler sequences which are not orbits, for example, it is impossible to have $a_n=a_{n+1}\neq a_{n+k}$ for some $k>1$.

This can be generalized to $a_n=a_{n+m}\neq a_{n+km},\; k>1$. In fact such sequences are not orbits of ANY function.

Other conditions can be obtained if $a_n$ has a limit, and tends to it with aproximately geometric speed. Then this limit must be an attracting point, and using the local linearization at this point one obtains VERY strong conditions on the sequence, and not only on its growth rate.

EDIT. For example, the following question was studied by Fatou. Instead of a rational function he considers an arbitrary analytic germ $f(z)=\lambda z+\ldots,\; |\lambda|<1$. He takes an orbit $a_n\to 0$, and makes a generating function $$g(\zeta)=\sum_{n=0}^\infty a_n\zeta^n.$$ He proves that $g$ is meromorphic in the whole plane with poles at the geometric progression $\lambda^{-k},\; k\geq 0$.

Fatou, P. Sur une classe remarquable de séries de Taylor. Ann. de l’Éc. Norm. (3) 27, 43-53 (1910).

There are many simpler sequences which are not orbits, for example, it is impossible to have $a_n=a_{n+1}\neq a_{n+k}$ for some $k>1$.

This can be generalized to $a_n=a_{n+m}\neq a_{n+km},\; k>1$. In fact such sequences are not orbits of ANY function.

Other conditions can be obtained if $a_n$ has a limit, and tends to it with aproximately geometric speed. Then this limit must be an attracting point, and using the local linearization at this point one obtains VERY strong conditions on the sequence.

There are many simpler sequences which are not orbits, for example, it is impossible to have $a_n=a_{n+1}\neq a_{n+k}$ for some $k>1$.

This can be generalized to $a_n=a_{n+m}\neq a_{n+km},\; k>1$. In fact such sequences are not orbits of ANY function.

Other conditions can be obtained if $a_n$ has a limit, and tends to it with aproximately geometric speed. Then this limit must be an attracting point, and using the local linearization at this point one obtains VERY strong conditions on the sequence.

EDIT. A related question was studied by Fatou. Instead of a rational function he considers a germ $f(z)=\lambda z+\ldots,\; |\lambda|<1$. He takes an orbit $a_n$ and makes a generating function $$g(\zeta)=\sum_{n=0}^\infty a_n\zeta^n.$$ He proves that $g$ is meromorphic in the whole plane with poles at the geometric progression $\lambda^{-k},\; k\geq 0$.

Fatou, P. Sur une classe remarquable de séries de Taylor. Ann. de l’Éc. Norm. (3) 27, 43-53 (1910).

There are many simpler sequences which are not orbits, for example, it is impossible to have $a_n=a_{n+1}\neq a_{n+k}$ for some $k>1$.

This can be generalized to $a_n=a_{n+m}\neq a_{n+km},\; k>1$. In fact such sequences are not orbits of ANY function.

There are many simpler sequences which are not orbits, for example, it is impossible to have $a_n=a_{n+1}\neq a_{n+k}$ for some $k>1$.

This can be generalized to $a_n=a_{n+m}\neq a_{n+km},\; k>1$. In fact such sequences are not orbits of ANY function.

Other conditions can be obtained if $a_n$ has a limit, and tends to it with aproximately geometric speed. Then this limit must be an attracting point, and using the local linearization at this point one obtains VERY strong conditions on the sequence.