Browsing by Author "Franco, Nuno"
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- Farey neighbors and hyperbolic Lorenz knotsPublication . Gomes, Paulo; Franco, Nuno; Silva, LuísBased on symbolic dynamics of Lorenz maps, we prove that, provided one conjecture due to Morton is true, then Lorenz knots associated to orbits of points in the renormalization intervals of Lorenz maps with reducible kneading invariant of type (X, Y)*S, where the sequences X and Y are Farey neighbors verifying some conditions, are hyperbolic.
- Genus and Braid Index Associated to Sequences of Renormalizable Lorenz MapsPublication . Franco, Nuno; Silva, LuisWe describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant (K(f)(-), = K(f)(+)) = (X, Y) * (S, W) in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from the ones corresponding to the factors of the *-product. Using this result we obtain explicit formulas for the genus and the braid index of this renormalizable Lorenz knots and links. Then we obtain explicit formulas for sequences of these invariants, associated to sequences of renormalizable Lorenz maps with kneading invariant (X, Y) * (S,W)*(n), concluding that both grow exponentially. This is specially relevant, since it is known that topological entropy is constant on the archipelagoes of renormalization.
- Genus for knots and links in renormalizable templates with several branch nodesPublication . Simões, Pedro; Silva, Luis; Franco, NunoWe apply kneading theory to describe the knots and links generated by the iteration of renormalizable nonautonomous dynamical systems with reducible kneading invariants, in terms of the links corresponding to each factor. As a consequence we obtain explicit formulas for the genus for this kind of knots and links.
- Partial classification of Lorenz knots: Syllable permutations of torus knots wordsPublication . Gomes, Paulo; Franco, Nuno; Silva, LuísWe define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots coded by some of these families of words are hyperbolic, by showing that they are neither satellites nor torus knots and making use of Thurston's theorem. Infinite families of hyperbolic Lorenz knots are generated in this way, to our knowledge, for the first time. The techniques used can be generalized to study other families of Lorenz knots.
- Symbolic dynamics and renormalization of non-autonomous k periodic dynamical systemsPublication . Franco, Nuno; Silva, Luis; Simões, PedroThe purpose of this paper was to introduce the symbolic formalism based on kneading theory, which allows us to study the renormalization of non-autonomous periodic dynamical systems.