Linear Response for a Family of Self-consistent Transfer Operators

Fanni M Sélley (1) , Matteo Tanzi (2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 Courant Inst. of Mathematical Sciences and NYU

We study a system of globally coupled uniformly expanding circle maps in the thermodynamic limit. The state of the system is described by a probability density and its evolution is given by the action of a nonlinear operator called the self-consistent transfer operator. Self-consistency is understood in the following sense: if ϕ corresponds to the system's state, then the evolution of ϕ is given by the application of the transfer operator of the circle map F ϕ. This is a ϕ−dependent map describing the dynamics of a single unit in the finite system where the interaction coming from all the nodes is replaced with the average of the interaction term with respect to the density ϕ. Assuming some level of smoothness of the coupled circle maps and of the coupling, we prove that when the coupling strength is sufficiently small, the system has a unique smooth stable state. We then show that this stable state is continuously differentiable as a function of the coupling strength. Finally, we prove that the derivative satisfies a linear response formula which at zero simplifies to a formula reminiscent to the one obtained for the linear transfer operators of perturbed expanding circle maps.

Domaines: Mathématiques [math]

Fanni M Sélley, Matteo Tanzi. Linear Response for a Family of Self-consistent Transfer Operators. Communications in Mathematical Physics, 2021, 382 (3), pp.1601-1624. ⟨10.1007/s00220-021-03983-6⟩. ⟨hal-03183609⟩