1. Show that if f R - R is continuous and has a 2-cycle (a,b}, then f has fixed point.
A map f : [1.7] [1,7] is defined so that f(1) = 4. f(2) = 7./(3) = 6,5(4) =
5. f(5) = 3. (f(6) = 2.f(7) = 1, and the corresponding points are joined so the map is
piecewise linear. Show that f has a 7-cycle but no 5-cycle.
3. If Fa(x) = 1 - to for I € R, show
i) Fx has fixed points for A -1/4.
ii) Fx has a 2-cycle for l 3/4.
iii) The 2-cycle is attracting for 3/45. If F(x) -
i) That F(x) has no points of period 1.
ii) Find the points of period 2.
iii) Graph this function to see that there are points of period 3.
iv) Does this contradict Sharkovsky's Theorem?
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