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On maximizing measures of homeomorphisms on compact manifolds

100%

Tal F., Addas-Zanata S.

Fundamenta Mathematicae

2008

tom 200

nr 2

145-159

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 $∫_X gdμ$, 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.

Persistence of fixed points under rigid perturbations of maps

100%

Addas-Zanata S., Salomão P.

Fundamenta Mathematicae

2014

tom 227

nr 1

1-19

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 $f̃_ϵ$ be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that $Fix(f̃_ϵ) = ∅$ for all ϵ > 0 sufficiently small. The proof follows from Kerékjártó's construction of Brouwer lines for orientation preserving homeomorphisms of the plane with no fixed points. This result turns out to be sharp with respect to the regularity assumption: there exists a diffeomorphism f with all the properties above, except that f is not real-analytic but only smooth, such that the above conclusion is false. Such a map is constructed via generating functions.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Addas-Zanata S.

1 Salomão P.

1 Tal F.

1 2014

1 2008

Wyniki wyszukiwania

On maximizing measures of homeomorphisms on compact manifolds

Persistence of fixed points under rigid perturbations of maps