Università degli Studi di Padova

“The topological entropy conjecture”

Martedì 11 Giugno 2024, ore 17:00 - Aula 1BC45 - Andrea Ulliana (Phd student Universtitat Zurich UZH)


In this talk I will give an introduction to the Topological Entropy Conjecture formulated by Shub, and highlight the most relevant related results from the litterature.

It is conjectured that the topological entropy of a diffeomorphism on a manifold can be controlled from below by the action induced by the diffeomorphism on the homology. The regularity of the diffeomorphism turns out to play a key role: the conjecture was disproved in the case of homeomorphisms, and conversely holds true if the diffeomorphism is $C^{\infty}$. For the finite regularity case, the problem is still open. We recently witnessed advancements, when some restrictions on the dynamics of the diffeomorphism were assumed.

Finally, we remark that this question led to the formulation of many other closely related problems, involving different global topological invariants.

Mathematics in Conversation