Let Eo and E₁ be the sets shown below, with certain points marked. Ео (0,0) (1/3, 1/9) E₁ (0,0) (2/3,0) (1,0) (1,0) (a) Write down contracting similarities Si, 1 ≤i≤4, on R² such that E₁ = US(E), marked points in E。 map to marked points in E₁ and S; (E) S; (E) is a single point for each i #j. (b) Sketch E2 U±1 S; (E₁). (c) Using results from Chapter 9 of Falconer, determine the Hausdorff and box dimensions of the attractor F of {S1, S2, S3, S4}. (d) Identify a set of seven contracting similarities Tį, 1 ≤ i ≤7, on R² such that the attractor of {T₁,..., T-} is equal to the set F defined in part (c). (e) Identify a set of five contracting similarities Ri, 1≤ i ≤ 5, on R² such that E₁ = UR(E) but such that the attractor of {R₁,..., Rs} is not equal to the set F defined in part (c), justifying your answer carefully.

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