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We study the notion of Fagnano orbits for dual polygonal billiards. We used them to characterize regular polygons and we study the iteration of the developing map.
Comment: 15 pages
Comment: 16 pages, 7 figures, added reference. Typos corrected and a forgotten term in de linearized system is added
We study the rate of growth of ratios of intervals delimited by the post-critical orbit of a map in the quasi-quadratic family $x\mapsto -|x|^\alpha +a.$ The critical order $\alpha$ is an arbitrary real number $\alpha>1.$ The range of the parameter $a$ is confined to an interval $(1,a_{\alpha})$ of length depending on the critical order. We prove that in every power-law family there is a unique parameter $p_{\alpha}$ corresponding to the kneading sequence $RLRRRLRC.$ Subsequently, we obtain monotonicity results concerning ratios of all intervals labeled by infinite post-critical orbit in the case of the kneading sequence $RLRL...$ This extends the results from \cite{P}, via refinement of the tools based on special properties of power-law mappings in non-euclidean metric.
Comment: 24 pages, 16 Figures, 8 Tables
We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C2-residual set of Hamiltonians for which every regular energy surface is either Anosov or it is in the closure of energy surfaces with zero Lyapunov exponents a.e. This is in the spirit of the Bochi-Mane dichotomy for area-preserving diffeomorphisms on compact surfaces and its continuous-time version for 3-dimensional volume-preserving flows.
In this announcement, we describe the solution in the C1 topology to a question asked by S. Smale on the genericity of trivial centralizers: the set of diffeomorphisms of a compact connected manifold with trivial centralizer residual in Diff^1 but does not contain an open and dense subset.
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