MathDB

Let

c\in\mathbb C

and A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}. a) Prove that for each

c\in C

A_c

is infinite. b) Prove that if

p\in A_1

, and p(z_0) \equal{} 0, then

|z_0| < 1.7

. c) Prove that each element of

A_c

is odd or even. Let f_c \equal{} z^2 \plus{} c\in \mathbb C[z]. We see easily that B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\} is a subset of

A_c

. Prove that in the following cases A_c \equal{} B_c. d)

|c| > 2

. e)

c\in \mathbb Q\backslash\mathbb Z

. f)

c

is a non-algebraic number g)

c

is a real number and c\not\in [ \minus{} 2,\frac14].

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