Periodic Reporting for period 2 - Emergence (Emergence of wild differentiable dynamical systems)
Período documentado: 2021-03-01 hasta 2022-08-31
Many physical or biological systems display time-dependent states which can be mathematically modelled by a differentiable dynamical system. The state of the system consists of a finite number of variables, and the short time evolution is given by a differentiable equation or the iteration of a differentiable map. The evolution of a state is called an orbit of the system. The theory of dynamical systems studies the long time evolution of the orbits.
For some systems, called chaotic, it is impossible to predict the state of an orbit after a long period of time. However, in some cases, one may predict the probability of an orbit to have a certain state. A paradigm is given by the Boltzmann ergodic hypothesis in thermodynamics: over long periods of time, the time spent by a typical orbit in some region of the phase space is proportional to the “measure” of this region. The concept of Ergodicity has been mathematically formalized by Birkhoff. Then it has been successfully applied (in particular) by the schools of Kolmogorov and Anosov in the USSR, and Smale in the USA to describe the statistical behaviours of typical orbits of many differentiable dynamical systems.
For some systems, called wild, infinitely many possible statistical behaviour coexist. Those are spread all over a huge space of different ergodic measures, as initially discovered by Newhouse in the 70's. Such systems are completely misunderstood. In 2016, contrarily to the general belief, it has been discovered that wild systems form a rather typical set of systems (in some categories).
This project proposes the first global, ergodic study of wild dynamics, by focusing on dynamics which are too complex to be well described by means of finitely many statistics, as recently quantified by the notion of Emergence. Paradigmatic examples will be investigated and shown to be typical in many senses and among many categories. They will be used to construct a theory on wild dynamics around the concept of Emergence.
For some systems, called chaotic, it is impossible to predict the state of an orbit after a long period of time. However, in some cases, one may predict the probability of an orbit to have a certain state. A paradigm is given by the Boltzmann ergodic hypothesis in thermodynamics: over long periods of time, the time spent by a typical orbit in some region of the phase space is proportional to the “measure” of this region. The concept of Ergodicity has been mathematically formalized by Birkhoff. Then it has been successfully applied (in particular) by the schools of Kolmogorov and Anosov in the USSR, and Smale in the USA to describe the statistical behaviours of typical orbits of many differentiable dynamical systems.
For some systems, called wild, infinitely many possible statistical behaviour coexist. Those are spread all over a huge space of different ergodic measures, as initially discovered by Newhouse in the 70's. Such systems are completely misunderstood. In 2016, contrarily to the general belief, it has been discovered that wild systems form a rather typical set of systems (in some categories).
This project proposes the first global, ergodic study of wild dynamics, by focusing on dynamics which are too complex to be well described by means of finitely many statistics, as recently quantified by the notion of Emergence. Paradigmatic examples will be investigated and shown to be typical in many senses and among many categories. They will be used to construct a theory on wild dynamics around the concept of Emergence.
Several results obtained during this program show the typicality of wild behavior.
- We recall that a point is periodic, if it goes back to it sent by the systems to its initial position after finite time called the period. The number of periodic points and its growth is used to measure how complicated is a dynamics. When the dynamical system is given by a polynomial, it can be shown that the number of periodic [isolated] points growth at most exponentially fast with the period. The famous mathematician Arnold ask in the early 90's wether it was also typically the case also for smooth dynamics. A [surprising] negative answer was bring by Pierre Berger, see Acta mathematica 2021. This implies
- A point of a system is stable if nearby points have close orbit. An important question is the wether the dynamics of such a point is simple. Pierre Berger and Sébastien Biebler showed it is in general not the case, even among invertible polynomial dynamics of the plan. Such points (and the points at their neighborhood) may have an orbit whose statistical behavior accumulates a huge set possibilities. This result was published at Journals of the American Mathematical Society and solved a 30 years old open problem.
- A natural questions is wether (smooth or polynomial) dynamics of the plan, which preserves the volume, can be embedded into a dynamics of a fluid of our (Euclidean) space. Pierre Berger, Anna Florio and Daniel Peralta-Salas that a generic steady ideal fluid flow of our space contain approximations of any volume preserving dynamics of the plan. This shows that the complexity of a steady ideal fluid flow of our space is (somehow) at least equal to those of volume preserving dynamics of the plan is dynamics. This solved a problem of Arnold from 1965.
- We recall that a point is periodic, if it goes back to it sent by the systems to its initial position after finite time called the period. The number of periodic points and its growth is used to measure how complicated is a dynamics. When the dynamical system is given by a polynomial, it can be shown that the number of periodic [isolated] points growth at most exponentially fast with the period. The famous mathematician Arnold ask in the early 90's wether it was also typically the case also for smooth dynamics. A [surprising] negative answer was bring by Pierre Berger, see Acta mathematica 2021. This implies
- A point of a system is stable if nearby points have close orbit. An important question is the wether the dynamics of such a point is simple. Pierre Berger and Sébastien Biebler showed it is in general not the case, even among invertible polynomial dynamics of the plan. Such points (and the points at their neighborhood) may have an orbit whose statistical behavior accumulates a huge set possibilities. This result was published at Journals of the American Mathematical Society and solved a 30 years old open problem.
- A natural questions is wether (smooth or polynomial) dynamics of the plan, which preserves the volume, can be embedded into a dynamics of a fluid of our (Euclidean) space. Pierre Berger, Anna Florio and Daniel Peralta-Salas that a generic steady ideal fluid flow of our space contain approximations of any volume preserving dynamics of the plan. This shows that the complexity of a steady ideal fluid flow of our space is (somehow) at least equal to those of volume preserving dynamics of the plan is dynamics. This solved a problem of Arnold from 1965.
- We expect to show that a typical dynamical systems displays a infinitely many statistical attractors, for a sense of typicality very strong.
- We expect also to initiate a theory by quantifying their difficulties to be described by a finite number of statistic.
- We expect also to initiate a theory by quantifying their difficulties to be described by a finite number of statistic.