Timeline for Growth of the size of iterated polynomials
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 19, 2015 at 16:51 | comment | added | Joe Silverman | The section in my book says that the limit $$\lim d^{-n}\log^+|p^n(a)|$$ converges, and differs from $\log^+|a|$ by a quantity that is bounded solely in terms of $p$. If you exponentiate that relation, you get the statement in my comment. | |
| Jul 19, 2015 at 15:40 | comment | added | Spock | @JoeSilverman Thanks a lot. Sorry that I am amateur but where can I find this result? I couldn't find them in the references in your answer. | |
| Jul 18, 2015 at 16:38 | vote | accept | Spock | ||
| Jul 15, 2015 at 22:42 | comment | added | Joe Silverman | @JoonasIlmavirta But one can say so much more. Specifically, there is a real number $H\ge1$ and positive constants $c_1$ and $c_2$, which depend on $p(x)$ and $a$, so that for all $n\ge1$ we have $c_1H^{d^n} \le |p^n(a)| \le c_2H^{d^n}$. | |
| Jul 15, 2015 at 22:34 | answer | added | Joe Silverman | timeline score: 8 | |
| Jul 15, 2015 at 20:44 | answer | added | Alexandre Eremenko | timeline score: 12 | |
| Jul 15, 2015 at 20:11 | comment | added | Joonas Ilmavirta | If $p$ has order $n$, there are constants $A$ and $B$ so that $-A+B^{-1}|x|^n\leq|p(x)|\leq A+B|x|^n$. If you can guarantee that the iterates (including the initial value) never reach zero, you can take $A=0$. Would such easy estimates be enough for you? | |
| Jul 15, 2015 at 20:01 | history | asked | Spock | CC BY-SA 3.0 |