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Let $P$ be a polynomial in complex variable $z$ of degree $d$

i.e. $P(z)= a_d z^d+.....+a_1 z+a_0$

Now I want to calculate following limit

$f(z) = \limsup_{n \to \infty} \frac{1}{d^n} (Log|P(z)^{*n} - b|)$

where $b \in$$\mathbb C$ is constant and $P(z)^{*n}$ is $n$ times composition of $P$.

I tried to take roots of $P^{*n}$ under $b$ and then root decomposition but couldn’t succeed. Any hint is welcome.

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  • $\begingroup$ the power $d^n$ in the denominator grows faster with $n$ than the logarithm in the numerator, so this limit will be zero. $\endgroup$ – Carlo Beenakker Nov 30 '18 at 11:37
  • $\begingroup$ For large values of $z$ above limit is approximately equal to leading term of $P(z)^{*n}$ so cannot be equal to zero. It depends on $z$. $\endgroup$ – Mayuresh L Nov 30 '18 at 12:05
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    $\begingroup$ you are right, I was confused. $\endgroup$ – Carlo Beenakker Nov 30 '18 at 12:07
  • $\begingroup$ Previously posted to m.se, math.stackexchange.com/questions/3010033/… $\endgroup$ – Gerry Myerson Nov 30 '18 at 21:17
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It is not quite clear what you mean by calculate. This limit is a familiar, classical object: it is called the Green function of the complement of the Julia set, or the equilibrium potential of the Julia set. It is positive and harmonic in the complement of the filled in Julia set of $P$, has asymptotics $\log|z|$ at infinity, and equals zero on the filled in Julia set (for all $b$ except possibly one). These properties define it uniquely. This limit is also subharmonic in the plane, and its Riesz measure is the measure of maximal entropy corresponding to $P$. It can be also described in terms of solution of the Bottcher equation at infinity. This function is one of the main tools of Holomorphic dynamics of polynomials.

For an introduction to the study of this object, see

MR0194595 Brolin, Hans Invariant sets under iteration of rational functions. Ark. Mat. 6 1965 103–144 (1965).

MR0762431 Douady, A.; Hubbard, J. H. Étude dynamique des polynômes complexes. Partie I. Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 84-2. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. 75 pp.

There are exactly two cases when this limit is an elementary function: when $P$ is a monomial and when $P$ is a Chebyshev polynomial. In all other cases one has only some infinite expression (a series, an infinite product etc.)

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