Consider the map F: x3x on the circle S¹ = R/Z. ⚫ Find all fixed points of this map. ⚫ Find all points of period 2. Put 10 = [0, 1), I₁ = [}, }), 1₂ = [3, 1). Let wx (x) be the index of the interval which contains F(x), F(x) = (x). Here and below F(x) = F(... F(x)...) The map which sends x to the sequence wo (x)w₁ (x) ... is called the fate map. k • Show that wk (F(x)) = @k+1(x). . Find the fates of all points of period 4, i.e. of all points such that F4(x) = x. For each of the points from the previous part, find its minimal period.