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The size of $|p^n(a)|$ can be very different depending on $a$. If $a$ is periodic (in particular, fixed) point of  $p$, this sequence will be bounded. If $a$ is in the domain of attraction of infinity then $\log|p^n(a)|$ is like $d^n$ where $d$ is the degree.

More precisely, for each polynomial there is a compact set $K(p)$ in the complex plane, called the filled in Julia set, such that for $a\in K(f)$ your sequence is bounded, while for $a\not\in K(p)$ it grows like $d^n$. When searching on Google look on "Holomorphic dynamics".

Remark. It is irrelevant for what I wrote above, whether $a$ is an integer or not and whether the coefficients are integer or complex. However, as Joe Silverman suggested in his comment, if the coefficients and $a$ are integers, then $p^n(a)$ are also integers, and as $K(f)$ may contain only finitely many integers, the sequence can be bounded only if it is pre-periodic, that is becomes periodic after the removal of finitely many terms.

EDIT. So for polynomials over integers and integer $a$, the question whether the sequence is bounded can be solved by finitely many arithmetic operations.

And from the discussion of the history I conclude that the result essentially follows from a theorem of L. E. Bottcher (early 1900-s) who proved the existence of an analytic function $\phi(z)\sim z$ near $\infty$ which conjugates the polynomial with $z^d$, that is $$\phi\circ p(z)=\phi(z)^d.$$ But Bottcher did not investigate the full domain of existence of this function, and $\phi$ may not exist in the whole domain of attraction of $\infty$. The idea to consider $\log|\phi|$ which is well-defined in the whole domain of attraction and can be extends to a subharmonic function in the whole plane was for the first time published by Brolin in 1965. Tate apparently had the same idea simultaneously or earlier but did not publish it.

The size of $|p^n(a)|$ can be very different depending on $a$. If $a$ is periodic point of  $p$, then this sequence will be bounded. If $a$ is in the domain of attraction of infinity then $\log|p^n(a)|$ is like $d^n$ where $d$ is the degree.

More precisely, for each polynomial there is a compact set $K(p)$ in the complex plane, called the filled in Julia set, such that for $a\in K(f)$ your sequence is bounded, while for $a\not\in K(p)$ it grows like $d^n$. When searching on Google look on "Holomorphic dynamics".

Remark. It is irrelevant for what I wrote above, whether $a$ is an integer or not and whether the coefficients are integer or complex. However, as Joe Silverman suggested in his comment, if the coefficients and $a$ are integers, then $p^n(a)$ are also integers, and as $K(f)$ may contain only finitely many integers, the sequence can be bounded only if it is pre-periodic, that is becomes periodic after the removal of finitely many terms.

EDIT. So for polynomials over integers and integer $a$, the question whether the sequence is bounded can be solved by finitely many arithmetic operations. But this is not completely trivial: one needs an explicit estimate, how many integer points can $K(f)$ contain, or something equivalent to this.

And from the discussion of the history I conclude that the result essentially follows from a theorem of L. E. Bottcher (early 1900-s) who proved the existence of an analytic function $\phi(z)\sim z$ near $\infty$ which conjugates the polynomial with $z^d$, that is $$\phi\circ p(z)=\phi(z)^d.$$ But Bottcher did not investigate the full domain of existence of this function, and $\phi$ may not exist in the whole domain of attraction of $\infty$. The idea to consider $\log|\phi|$ which is well-defined in the whole domain of attraction and can be extended to a subharmonic function in the whole plane was for the first time published by Brolin in 1965. Tate apparently had the same idea simultaneously or earlier but did not publish it.

The size of $|p^n(a)|$ can be very different depending on $a$. If $a$ is periodic (in particular, fixed) point of $p$, this sequence will be bounded. If $a$ is in the domain of attraction of infinity then $\log|p^n(a)|$ is like $d^n$ where $d$ is the degree.

More precisely, for each polynomial there is a compact set $K(p)$ in the complex plane, called the filled in Julia set, such that for $a\in K(f)$ your sequence is bounded, while for $a\not\in K(p)$ it grows like $d^n$. When searching on Google look on "Holomorphic dynamics".

Remark. It is irrelevant for what I wrote above, whether $a$ is an integer or not and whether the coefficients are integer or complex. However, as Joe Silverman suggested in his comment, if the coefficients and $a$ are integers, then $p^n(a)$ are also integers, and as $K(f)$ may contain only finitely many integers, the sequence can be bounded only if it is pre-periodic, that is becomes periodic after the removal of finitely many terms.

EDIT. So for polynomials over integers and integer $a$, the question whether the sequence is bounded can be solved by finitely many arithmetic operations.

And from the discussion of the history I conclude that the result essentially follows from a theorem of L. E. Bottcher (early 1900-s) who proved the existence of an analytic function $\phi(z)\sim z$ near $\infty$ which conjugates the polynomial with $z^d$, that is $$\phi\circ p(z)=\phi(z)^d.$$ But Bottcher did not investigate the full domain of existence of this function, and $\phi$ may not exist in the whole domain of attraction of $\infty$. The idea to consider $\log|\phi|$ which is well-defined in the whole domain of attraction and can be extends to a subharmonic function in the whole plane was for the first time published by Brolin in 1965. Tate apparently had the same idea simultaneously or earlier but did not publish it.

The size of $|p^n(a)|$ can be very different depending on $a$. If $a$ is periodic (in particular, fixed) point of $p$, this sequence will be bounded. If $a$ is in the domain of attraction of infinity then $\log|p^n(a)|$ is like $d^n$ where $d$ is the degree.

More precisely, for each polynomial there is a compact set $K(p)$ in the complex plane, called the filled in Julia set, such that for $a\in K(f)$ your sequence is bounded, while for $a\not\in K(p)$ it grows like $d^n$. When searching on Google look on "Holomorphic dynamics".

Remark. It is irrelevant for what I wrote above, whether $a$ is an integer or not and whether the coefficients are integer or complex. However, as Joe Silverman suggested in his comment, if the coefficients and $a$ are integers, then $p^n(a)$ are also integers, and as $K(f)$ may contain only finitely many integers, the sequence can be bounded only if it is pre-periodic, that is becomes periodic after the removal of finitely many terms.

EDIT. So for polynomials over integers and integer $a$, the question whether the sequence is bounded can be solved by finitely many arithmetic operations.

And from the discussion of the history I conclude that the result essentially follows from a theorem of L. E. Bottcher (early 1900-s) who proved the existence of an analytic function $\phi(z)\sim z$ near $\infty$ which conjugates the polynomial with $z^d$, that is $$\phi\circ p(z)=\phi(z)^d.$$ But Bottcher did not investigate the full domain of existence of this function, and $\phi$ may not exist in the whole domain of attraction of $\infty$. The idea to consider $\log|\phi|$ which is well-defined in the whole domain of attraction and can be extends to a subharmonic function in the whole plane was for the first time published by Brolin in 1965. Tate apparently had the same idea simultaneously or earlier but did not publish it.

The size of $|p^n(a)|$ can be very different depending on $a$. If $a$ is periodic (in particular, fixed) point of $p$, this sequence will be bounded. If $a$ is in the domain of attraction of infinity then $\log|p^n(a)|$ is like $d^n$ where $d$ is the degree.

More precisely, for each polynomial there is a compact set $K(p)$ in the complex plane, called the filled in Julia set, such that for $a\in K(f)$ your sequence is bounded, while for $a\not\in K(p)$ it grows like $d^n$. When searching on Google look on "Holomorphic dynamics".

Remark. It is irrelevant for what I wrote above, whether $a$ is an integer or not and whether the coefficients are integer or complex. However, as Joe Silverman suggested in his comment, if the coefficients and $a$ are integers, then $p^n(a)$ are also integers, and as $K(f)$ may contain only finitely many integers, the sequence can be bounded only if it is pre-periodic, that is becomes periodic after the removal of finitely many terms.

The size of $|p^n(a)|$ can be very different depending on $a$. If $a$ is periodic (in particular, fixed) point of $p$, this sequence will be bounded. If $a$ is in the domain of attraction of infinity then $\log|p^n(a)|$ is like $d^n$ where $d$ is the degree.

More precisely, for each polynomial there is a compact set $K(p)$ in the complex plane, called the filled in Julia set, such that for $a\in K(f)$ your sequence is bounded, while for $a\not\in K(p)$ it grows like $d^n$. When searching on Google look on "Holomorphic dynamics".

Remark. It is irrelevant for what I wrote above, whether $a$ is an integer or not and whether the coefficients are integer or complex. However, as Joe Silverman suggested in his comment, if the coefficients and $a$ are integers, then $p^n(a)$ are also integers, and as $K(f)$ may contain only finitely many integers, the sequence can be bounded only if it is pre-periodic, that is becomes periodic after the removal of finitely many terms.

EDIT. So for polynomials over integers and integer $a$, the question whether the sequence is bounded can be solved by finitely many arithmetic operations.

And from the discussion of the history I conclude that the result essentially follows from a theorem of L. E. Bottcher (early 1900-s) who proved the existence of an analytic function $\phi(z)\sim z$ near $\infty$ which conjugates the polynomial with $z^d$, that is $$\phi\circ p(z)=\phi(z)^d.$$ But Bottcher did not investigate the full domain of existence of this function, and $\phi$ may not exist in the whole domain of attraction of $\infty$. The idea to consider $\log|\phi|$ which is well-defined in the whole domain of attraction and can be extends to a subharmonic function in the whole plane was for the first time published by Brolin in 1965. Tate apparently had the same idea simultaneously or earlier but did not publish it.