We prove that given a compact n-dimensional connected Riemannian manifold X and a continuous function g: X → ℝ, there exists a dense subset of the space of homeomorphisms of X such that for all T in this subset, the integral , considered as a function on the space of all T-invariant Borel probability measures μ, attains its maximum on a measure supported on a periodic orbit.
Let f: S¹ × [0,1] → S¹ × [0,1] be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift f̃: ℝ × [0,1] → ℝ × [0,1] we have Fix(f̃) = ℝ × 0 and that f̃ positively translates points in ℝ × 1. Let be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that for all ϵ > 0 sufficiently small. The proof follows from Kerékjártó’s construction of Brouwer lines for orientation preserving homeomorphisms...
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