Dynamics of surface maps

In the sciences in general, the phrase “route to chaos” has come to refer to a metaphor when some physical, biological, economic, or social system transitions from one exhibiting order to one displaying randomness (or chaos). The goal of this project is to understand which universal mechanisms explain that transition, and how one can describe systems that operate in a region between order and complete chaos, where the complexity is maximal. In other words, the goal is to understand the processes by which a system evolves from periodic toward chaotic behavior as one or more parameters governing the behavior of the system are varied. This has only been understood in low dimensional dynamics (which are as simple as a system involving one variable). The present project presents a novel approach that allows us to move away from those limitations. Moreover, the project brings some perspectives to new advances in mathematics and science in general, not necessarily in terms of bringing techniques or theorems, but in terms of ideas and conceptual approaches. The project will also support the training of graduate students, directly involved in this research.

The current project presents a tentative global framework toward describing a large class of area contracting surface dynamics of the disc with zero and low entropy, inspired partially by the developments in the one-dimensional theory of interval maps. More precisely, the project proposes a class of intermediate smooth dynamics between one and higher dimensions (that includes the Henon family with Jacobian smaller than 1 /4 ) where it could be possible to develop a similar one-dimensional type approach that could help to understand dynamics with zero or low topological entropy. In particular, it presents a general framework to understand the transition from zero entropy to positive entropy.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.