We develop a general theory for the dynamics of the geodesic flow of locally CAT(k) branched coverings and we show that it does not depend on the specific covering but only on the base space, the branching set and the degree of the covering. We find an explicit formula for the entropy: it equals the topological pressure of the associated natural dynamical system on the space of broken geodesics with a natural geometric potential. We find the exact asymptotic of the entropy as the number of sheets of the branched covering goes to infinity, as well as asymptotic properties of the measures of maximal entropy.
We give a short proof of the following extension of a recent result of Gadre, Maher, Pfaff and Uyanki: Consider a closed hyperbolic 3-manifold M which fibers over the circle, with fiber a closed surface S. Any measure class on the ideal boundary of hyperbolic 3-space which is invariant and totally ergodic under the act...
Consider a holomorphic family (fλ)λ∈Λ$(f_\lambda)_{\lambda \in \Lambda }$ of polynomial maps on C$\mathbb {C}$ with the property that a critical point of fλ$f_\lambda$ is persistently preperiodic to a repelling periodic point of fλ$f_\lambda$ . Let Ω$\Omega$ be a bounded stable component of Λ$\Lambda$ with the property...
F. Bianchi, Y. He· Journal of the London Mathem...· 2 citations
We address a question of Canary--Taylor and Canary--Minsky--Taylor for sufficiently symmetric compact hyperbolic $3$-orbifolds with totally geodesic boundary. We give sufficient conditions on the symmetries of such an orbifold to show that, among the complete convex cocompact structures on the interior, the structure w...
Sami Douba, Andr'es Sambarino, Franco Vargas Pallete· 0 citations
We prove that the space of isometry classes of nonempty compact metric spaces, equipped with the Gromov--Hausdorff distance, is homeomorphic to the real separable infinite-dimensional Hilbert space. We construct a continuous assignment of full-support probability measures that is equivariant under isometries and finite...
We prove the unboundedness of the spectral distance on the space of Lagrangians Hamiltonian isotopic to the zero section within the disc cotangent bundle of a closed manifold satisfying some topological assumptions. This disproves the generalization of a conjecture due to Claude Viterbo. The proof relies on a direct co...
Filip Broćić, Baptiste Serraille, Ibrahim Trifa· 0 citations
We study the (typical) singularities of the distance function from a point in the ambient space to a polytopal complex in Euclidean space. By means of the Goresky-MacPherson splitting we generalize the existing linear theory to the quadratic distance function. Critical points occur on `selected'strata of the PL-space,...
D. Siersma· 0 citations
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