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On maximizing measures of homeomorphisms on compact manifolds

Fábio Armando Tal;  Salvador Addas-Zanata — 2008

Fundamenta Mathematicae

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 g d μ , 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.

Density of the set of symbolic dynamics with all ergodic measures supported on periodic orbits

Tatiane Cardoso Batista;  Juliano dos Santos Gonschorowski;  Fabio Armando Tal — 2015

Fundamenta Mathematicae

Let K be the Cantor set. We prove that arbitrarily close to a homeomorphism T: K → K there exists a homeomorphism T̃: K → K such that the ω-limit of every orbit is a periodic orbit. We also prove that arbitrarily close to an endomorphism T: K → K there exists an endomorphism T̃: K → K with every orbit finally periodic.

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