The Palis Conjecture describes the behavior of smooth dynamical systems. It asserts that the chaos observed in typical systems can be successfully described with probabilistic methods. Only for one-dimensional systems has this conjecture been proved. The long-term goal of this project is to prove the conjecture for the Henon family, which consists of two-dimensional systems. The essential part of a system is concentrated in its attractor. In particular, the probabilistic behavior of a system is closely related to the microscopic geometric properties of the attractor. Renormalization is a method to study this microscopic geometry. A central theme in dynamics is the exploration of renormalization beyond the theory of one-dimensional systems, where it has been instrumental in making progress. This project concentrates on the development of a renormalization theory for Henon maps. Renormalization has already been successful in improving the understanding of Henon maps that are at transition to chaos, dissipative maps at the accumulation of period-doubling-type. The immediate goals of the proposed research are the following: first, to extend the period-doubling renormalization theory to more general combinatorial types in the Henon family; second, to refine the theory for the dynamics of Henon maps at the accumulation of period doubling; and finally, to use the renormalization results to establish the Palis Conjecture at the accumulation of period doubling.
Dynamics is the study of processes. This can mean processes generated by mechanical systems (e.g., the solar system), but also chemical, biological, or even sociological processes. An underlying idea is that the chaos observed in such systems can be understood qualitatively with the aid of a limited number of mechanisms. One goal of dynamics is to help engineers and scientists by explaining the mechanisms that are at play in their chaotic systems. Indeed, dynamics has led to many industrial applications. The processes dynamics studies are deterministic. Thus, if the positions and speeds of all the planets in the solar system were known now, dynamical theory could predict with great precision what those quantities will be next month. The future is, in principle, determined. Unfortunately, when a system is chaotic it becomes practically impossible to predict its future precisely. During the last fifty years it has become clear that, in order to describe deterministic chaos, science needs probabilistic methods. It has also become clear that chaos is in a quite real sense "very well organized." This organization is reflected in its probabilistic behavior. We all have seen the "bell curve" being used in many applications. It is always the same bell curve, an aspect of the organization within chaos. The relevant probabilistic laws are the phenomena observed in chaos. The explanation of these laws is intrinsically related to the microscopic properties of what scientists know as "attractors," which contain the essential aspects of the observed behavior of systems. Renormalization is a method to study this microscopic geometry. The organization of chaos becomes clearly visible exactly at this microscopic scale. Dynamics is very far from a complete understanding of chaos, but renormalization has been instrumental in the most sophisticated theories available at the present time. This project will explore further applications of renormalization. In particular, it will concentrate on systems that are related to the creation of chaos, the so-called Henon systems.
Dynamics studies time evolution of systems. The areas of application range from the solar system, global weather on earth, to the future and past of the cosmos. And everything between. Although the enormous variety of systems, one often observes common features. The role of mathematics in dynamics can be found in fundamental contributions to geometrical and probabilistic aspects of dynamics but also in concrete algorithms for predictions. We are still very far from a global understanding of dynamics. Generally speaking we only understand the dynamics of real world systems by numerical simulations. Refined theories are only available in specific and relatively simple systems. The degree of complexity one can expect increases with the number of observables involved, the dimension of the system. Among the most refined theories is one-dimensional dynamics. Although one-dimensional dynamics is the simplest of the simplest, the one-dimensional theory is extremely rich. The central tool in one-dimensional dynamics is renormalization. It connects the combinatorial structure of the systems with its geometrical and its probabilistic behavior. One step up, the theory for two-dimensional dynamics is still wide open. The main results of this proposal concern the transition to chaos in two-dimensional systems. Intellectual merit of the proposed activity. Cyclic behavior is the simplest type of dynamics. The system evolves and eventually it is back in the same state it started in. Usually, this cyclic behavior is stable. Small changes along the way will not dramatically change the cyclic behavior. Chaotic behavior charaterises itself by great sensitivity. In such systems cyclic behavior is not prevalent. The smallest changes along the way can change the behavior dramatically. A relevant consideration is that even small amounts of added CO2 to the atmosphere might have dramatic consequences on the weather. We are very far from a general understanding of chaotic dynamics. However, the transition from cyclic to chaotic dynamics is better understood. In particular, there is a very rich theory describing transition to chaos, the so-called period doubling route to chaos. The system is still being dominated by cyclic behavior but as it gets closer to transition, the boundary of chaos, the period of cyclicity doubles repeatedly. Eventually, there is no prevalent cyclic behavior anymore and the system is chaotic. A system consists of a state space and an evolultion-rule explaning how the state of the system changes when time passes. Cyclic behavior corresponds to a loop in this space.The system returns after a while where it started: a loop of states. Chaotic behavior corresponds to orbits which are much more compliacted and are not loops. The geometry of these orbits plays a crucial role in dynamics. For example, the geometrical properties of orbits are related to how transitions occur but also to the probabilistic behavior of the system. Behavior, geometry and probability are close related. The integrating tool is renormalization. A deep discovery by Coullet-Tresse/Feigenbaum in the seventies of the last centrury, revealed that the geometry of orbits at transition to chaos is universal. It does not depend on the specific properties of the system. Period doubling routes to chaos occur always with the exactly the same geometrical features. This universality has been rigourously understood in one-dimensional dynamics and moreover the one-dimensional geometrical structures has been measured in many real world systems. The Conjecture was that indeed, there is always one-dimensional universal geometry associated to period doibling routes to chaos. (The Conjecture apllies to dissipative systems with only one degree of instability.) Henon maps play a crucial role in understanding transistions observed in general smooth dynamics. In particular, they play a crucial role in the transition to chaos in dissipative dynamics. The main result of this project says that unfortuantely, the geometry at transition to chaos for two-dimensional Henon maps is not the universal geometrical from one-dimensional dynamics. The situation is much more delicate. Part of the one-dimensional features break down but from a probabilitic point of view the probability of geometrical break down is zero. That's why this geometrical breakdown is very hard to detect in real world systems. The main result from this project is that universilty in higher dimensional dynamics has a probabilitic nature: Probabilistic Universality. Broader impacts resulting from the proposed activity. The proposed activity will result in deeper insights into small scale structure of dynamical systems, in training of highly qualified graduate students and post-docs who will apply their skills in academia and industry, in broader interactions between experts in various branches of dynamics.