We consider the Lagrange and the Markov dynamical spectra associated to horseshoes on a surface with Hausdorff dimension greater than one. We show that for a ‘large’ set of real functions on the surface and for ‘typical’ horseshoes with Hausdorff dimension greater than one, both the Lagrange and the Markov dynamical spectra have persistently non-empty interior.

We consider the Lagrange and the Markov dynamical spectra associated with the geodesic flow on surfaces of negative curvature. We show that for a large set of real functions on the unit tangent bundle and typical metrics with negative curvature and finite volume, both the Lagrange and the Markov dynamical spectra have non-empty interiors.

We consider the Lagrange and the Markov dynamical spectra associated with a conservative Anosov flow on a compact manifold of dimension 3 (including geodesic flows of negative curvature and suspension flows). We show that for a large set of real functions and typical conservative Anosov flows, both the Lagrange and Markov dynamical spectra have a non-empty interior.

Let N be an n-dimensional compact riemannian manifold, with n ≥ 2. In this paper, we prove that for any α ϵ 2 [0, n], the set consisting of homeomorphisms on N with lower and upper metric mean dimensions equal to α is dense in Hom(N). More generally, given α, β ϵ [0; n], with α ≤ β , we show the set consisting of homeomorphisms on N with lower metric mean dimension equal to α and upper metric mean dimension equal to β is dense in Hom(N). Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to n is residual in Hom(N).

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