Speaker: Eran Igra
Abstract: Assume we have a surface diffeomorphism which can be deformed by some relative isotopy to a Pseudo-Anosov map. It is well-known that as we return along the isotopy to the original map all the periodic orbits for the Pseudo-Anosov map persist – or in other words, the topology forces the existence of complex dynamics which cannot be removed under relative isotopies (in particular, the Pseudo-Anosov map is “dynamically minimal”). Now, let us replace our surface diffeomorphism with a smooth flow on a three-dimensional manifold – then, can we find some topological constrain on the flow which forces complex dynamics to appear? For example, a constrain which forces the existence of a strange chaotic attractor?
In this talk we give a partial answer to this question. Inspired by the Thurston-Nielsen Classification Theorem and the Orbit Index Theory we prove that in certain heteroclinic scenarios one can define a class of periodic orbits for the flow which persist – without changing their knot type – under a certain class of smooth homotopies of the vector field which keep the heteroclinic condition fixed. This has the following meaning: assume we can smoothly deform the dynamics trapped between a heteroclinic knot into a hyperbolic (or more precisely, singular hyperbolic) dynamical system, then all the periodic orbits for the singular hyperbolic system are also generated by the original flow. Following that will show how our results can be applied to study the dynamics of the Lorenz and Rössler attractors, and time permitting, we will conjecture how our can be generalized to derive a forcing theory for three-dimensional flows.
Time: 4:00~5:00 pm, Tuesday, December 24, 2024
Location: Room 1610 at SIMIS, Block A, No. 657 Songhu Road, Yangpu District, Shanghai
English 
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