Periodic Reporting for period 4 - SOS (Smooth dynamics via Operators, with Singularities)
Periodo di rendicontazione: 2023-03-01 al 2023-12-31
The statistical study (ergodic theory) of smooth dynamical systems has seen important progress since the beginning of the twenty first century, in part due to the development of a new technical tool of "anisotropic Banach" spaces. Recently, such tools have yielded exponential mixing for the 2D periodic Lorentz gas (flow), which is the archetypal smooth system with singularities, and is motivated by statistical physics.
The project "Smooth dynamics via Operators, with Singularities" develops mathematical analysis tools to describe this and other dynamical systems originating from natural physical models. Such systems were up to now not approachable by rigorous methods, due to the presence of singularities. Objectives include a more precise description of the long term behaviour of the systems, and how they react to small changes of parameters ("linear response"). Potential applications include climate modelling.
The main conclusions of the action are twofold:
1) In the presence of singularities, the measure of maximal entropy (MME, as oppoosed to the physical measure) of hyperbolic systems behaves significantly differently as for smooth dynamical systems. Existence is not guaranteed unconditionally, and the speed of mixing is not necessarily exponential. This has been established in the setting of dispersive billiards (the 2D periodic Lorentz gas map or flow), using transfer operators.
2) In the presence of bifurcations, response is often fractional and not linear. In addition to the transfer operators, probabilistic techniques are needed to formulate and prove appropriate results. This has been established for the model of the logistic (quadratic) family
The project "Smooth dynamics via Operators, with Singularities" develops mathematical analysis tools to describe this and other dynamical systems originating from natural physical models. Such systems were up to now not approachable by rigorous methods, due to the presence of singularities. Objectives include a more precise description of the long term behaviour of the systems, and how they react to small changes of parameters ("linear response"). Potential applications include climate modelling.
The main conclusions of the action are twofold:
1) In the presence of singularities, the measure of maximal entropy (MME, as oppoosed to the physical measure) of hyperbolic systems behaves significantly differently as for smooth dynamical systems. Existence is not guaranteed unconditionally, and the speed of mixing is not necessarily exponential. This has been established in the setting of dispersive billiards (the 2D periodic Lorentz gas map or flow), using transfer operators.
2) In the presence of bifurcations, response is often fractional and not linear. In addition to the transfer operators, probabilistic techniques are needed to formulate and prove appropriate results. This has been established for the model of the logistic (quadratic) family
The PI has proved, with Mark Demers, existence and mixing of the measure of maximal entropy of the map associated to the 2D periodic Lorentz gas.
Still with Mark Demers, the PI studied the thermodynamic formalism for the 2D periodic Lorentz gas, for a family of potential related to the unstable Jacobian. Then, with the PhD student Jérôme Carrand and Mark Demers, the PI proved existence and mixing of the measure of maximal entropy of the flow associated to the 2D periodic Lorentz gas. This uses a study by the second PhD student Jérôme Carrand of thermodynamic formalism for the 2D periodic Lorentz gas, for a family of potential related to the first collision time.
The first PhD Student, Malo Jézéquel, has been able to control the growth of dynamical determinants and zeta functions of differentiable (non analytic) maps and flows, with applications to the global trace formula. (Revisiting the Milnor--hurston kneading theory to obtain nuclear power decompositions in the non-analytic setting.)
With a Postdoc (Juho Leppänen), the PI has studied fractional response and fractional susceptibility function for transversal families of piecewise expanding maps.
With Daniel Smania, the PI has obtained pioneering results on the fractional susceptibility function for the quadratic family. With Magnus Aspenberg and Tomas Persson, the PI has proved an almost sure invariance principle for the quadratic family
The postdocs (Leppänen, Sedro, Selley, Castorrini, Korepanov, and Wormell) obtained results on central limit theorems with a rate of convergence for time-dependent intermittent maps, quadratic response, and differentiability and non-monotonicity of the rotation number, conditional mixing in deterministic chaos, speed of mixing for the measure of maximal entropy of Sinai billiards, .
36 articles have been published and 5 were submitted. One workshop, one conference, one school, and one smaller meeting were organised.
Still with Mark Demers, the PI studied the thermodynamic formalism for the 2D periodic Lorentz gas, for a family of potential related to the unstable Jacobian. Then, with the PhD student Jérôme Carrand and Mark Demers, the PI proved existence and mixing of the measure of maximal entropy of the flow associated to the 2D periodic Lorentz gas. This uses a study by the second PhD student Jérôme Carrand of thermodynamic formalism for the 2D periodic Lorentz gas, for a family of potential related to the first collision time.
The first PhD Student, Malo Jézéquel, has been able to control the growth of dynamical determinants and zeta functions of differentiable (non analytic) maps and flows, with applications to the global trace formula. (Revisiting the Milnor--hurston kneading theory to obtain nuclear power decompositions in the non-analytic setting.)
With a Postdoc (Juho Leppänen), the PI has studied fractional response and fractional susceptibility function for transversal families of piecewise expanding maps.
With Daniel Smania, the PI has obtained pioneering results on the fractional susceptibility function for the quadratic family. With Magnus Aspenberg and Tomas Persson, the PI has proved an almost sure invariance principle for the quadratic family
The postdocs (Leppänen, Sedro, Selley, Castorrini, Korepanov, and Wormell) obtained results on central limit theorems with a rate of convergence for time-dependent intermittent maps, quadratic response, and differentiability and non-monotonicity of the rotation number, conditional mixing in deterministic chaos, speed of mixing for the measure of maximal entropy of Sinai billiards, .
36 articles have been published and 5 were submitted. One workshop, one conference, one school, and one smaller meeting were organised.
Further goals of the project include:
The measure of maximal entropy for the Sinai billiard flow.
Further statistical properties of the measure of maximal entropy for the Sinai billiard map.
Intrinsic resonances of the 2D periodic Lorentz gas via the dynamical zeta function.
Fine statistical properties of (infinite measure) semi-dispersing billiards with non compact cusps.
Limit theorems for fractional response for transversal families of smooth nonuniformly hyperbolic maps (including the logistic family).
The measure of maximal entropy for the Sinai billiard flow.
Further statistical properties of the measure of maximal entropy for the Sinai billiard map.
Intrinsic resonances of the 2D periodic Lorentz gas via the dynamical zeta function.
Fine statistical properties of (infinite measure) semi-dispersing billiards with non compact cusps.
Limit theorems for fractional response for transversal families of smooth nonuniformly hyperbolic maps (including the logistic family).