MathDB

Periodic points

Source: Brazilian Math Olympiad, 2004

October 19, 2004

algebra unsolvedalgebra

Problem Statement

Let

a

and

b

be real numbers. Define

f_{a,b}\colon R^2\to R^2

by

f_{a,b}(x;y)=(a-by-x^2;x)

. If

P=(x;y)\in R^2

, define

f^0_{a,b}(P) = P

and

f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))

for all nonnegative integers

k

. The set

per(a;b)

of the periodic points of

f_{a,b}

is the set of points

P\in R^2

such that

f_{a,b}^n(P) = P

for some positive integer

n

. Fix

b

. Prove that the set

A_b=\{a\in R \mid per(a;b)\neq \emptyset\}

admits a minimum. Find this minimum.

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