Linear Response for a Family of Self-Consistent Transfer Operators

Dynamics of SIR model with vaccination and heterogeneous behavioral response of individuals modeled by the Preisach operator

We study global dynamics of an SIR model with vaccination, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. We present a condition for the global stability of the infection-free equilibrium state. If this condition does not hold true, the model has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals. In this case, we show that every trajectory converges either to an endemic equilibrium or to a periodic orbit. Under addi…

间歇性圆映射的线性响应

Linear Response for Intermittent Circle Maps

本文通过锥技术研究间歇性圆映射的SRB测度弱微分性，证明线性响应并展示其统计稳定性。

Frequency Response of Uncertain Systems: Strong Kharitonov-Like Results

In this paper, we study the frequency response of uncertain systems using Kharitonov stability theory on first order complex polynomial set. For an interval transfer function, we show that the minimal real part of the frequency response at any fixed frequency is attained at some prescribed vertex transfer functions. By further geometric and algebraic analysis, we identify an index for strict positive realness of interval transfer functions. Some extensions and applications in positivity verification and robust absolute stability of feedback control systems are also presented.

关于不连续扰动下光滑映射线性响应的研究

On linear response for discontinuous perturbations of smooth endomorphisms

本文研究了不连续扰动下光滑映射的线性响应，证明了在满足均匀混合条件时，物理测度对参数具有Lipschitz连续性，并讨论了抽象假设在具体例子中的应用。

Regularity of characteristic exponents and linear response for transfer operator cocycles

We consider cocycles obtained by composing sequences of transfer operators with positive weights, associated with uniformly expanding maps (possibly having countably many branches) and depending upon parameters. Assuming $C^k$ regularity with respect to coordinates and parameters, we show that when the sequence is picked within a certain uniform family the top characteristic exponent and generator of top Oseledets space of the cocycle are $C^{k-1}$ in parameters. As applications, we obtain a linear response formula for the equivariant measure associated with random products of uniformly expand…

Functional analysis behind a Family of Multidimensional Continued Fractions: Part II

This paper is a direct continuation of "Functional analysis behind a Family of Multidimensional Continued Fractions: Part I," in which we started the exploration of the functional analysis behind the transfer operators for triangle partition maps, a family that includes many, if not most, well-known multidimensional continued fraction algorithms. This allows us now to find eigenfunctions of eigenvalue 1 for transfer operators associated with select triangle partition maps on specified Banach spaces. We proceed to prove that the transfer operators, viewed as acting on one-dimensional families o…

Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

We characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

Self consistent transfer operators. Invariant measures, convergence to equilibrium, linear response and control of the statistical properties

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented in the paper are valid in a weak coupling regime, the existence results for the …

On the leading eigenvalue of transfer operators of the Farey map with real temperature

We study the spectral properties of a family of generalized transfer operators associated to the Farey map. We show that when acting on a suitable space of holomorphic functions, the operators are self-adjoint and the positive dominant eigenvalue can be approximated by means of the matrix expression of the operators.

Spectral analysis of transfer operators associated to Farey fractions

The spectrum of a one-parameter family of signed transfer operators associated to the Farey map is studied in detail. We show that when acting on a suitable Hilbert space of analytic functions they are self-adjoint and exhibit absolutely continuous spectrum and no non-zero point spectrum. Polynomial eigenfunctions when the parameter is a negative half-integer are also discussed.