Dynamical Systems seminar: Robust transitivity: from uniform hyperbolicity, through blenders, and to computer-assisted proofs

Note – the talk will be online, on the Microsoft Teams app.

Speaker: Marisa Dos Reis Cantarino (Monash University)

Abstract: Blenders originally emerged as objects in dynamical systems as an example given by C. Bonatti and L. Díaz of a system which is not uniformly hyperbolic but it is robustly transitive. This means that most points in the system visit every region of it in the future, and this behaviour is robust. Before the introduction of blenders, robust transitivity was known to hold for other particular cases.

Roughly speaking, on an n-manifold, a blender is a hyperbolic invariant subset of the system that allows for robust intersections of s-dimensional stable manifolds and u-dimensional unstable manifolds, with s + u < n. The way to make this intersection robust without “the right dimensions” is to make the stable set of the blender to “fill the space as if it is higher dimensional”. This intersection allows the existence of robust heterodimensional cycles, giving conditions for robust transitivity.

All these concepts and a short overview of the history of this problem will be introduced via examples. We will then present a family of D.A. (derived-from-Anosov) systems on the 3-torus for which we are exploring robust transitivity using the presence of blenders. The existence of a blender and robust transitivity are proved using computer-assisted strategies, in particular interval arithmetics. This is ongoing work made with the collaboration of Andy Hammerlindl and Warwick Tucker.

Time: Thur., Oct. 31, 2024
Location: Microsoft Teams Meeting ID: 954 585 487 551 9 (Passcode: tzAM9w)

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