Related papers: Linear response due to singularities
In this note we consider a tent-like family with a cusp at the singular point and show that the linear response holds for certain perturbations of this family. This contrasts the tent-like maps with finite derivatives at the singularity.…
We consider a two-parameter family of maps $T_{\alpha, \beta}: [0,1] \to [0,1]$ with a neutral fixed point and a non-flat critical point. Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables…
We show a linear response statement for fixed points of a family of Markov operators which are perturbations of mixing and regularizing operators. We apply the statement to random dynamical systems on the interval given by a deterministic…
Consider a smooth one-parameter family t -> f_t of dynamical systems f_t, with |t|<epsilon. Assume that for all t (or for many t close to t=0) the map f_t admits a unique SRB invariant probability measure m_t. We say that linear response}…
We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and,…
We consider a family of Pomeau-Manneville type interval maps $T_\alpha$, parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$.…
Understanding the linear response of any system is the first step towards analyzing its linear and nonlinear dynamics, stability properties, as well as its behavior in the presence of noise. In non-Hermitian Hamiltonian systems, calculating…
We study differentiability properties in a particular case of the Palmer's linearization Theorem, which states the existence of an homeomorphism $H$ between the solutions of a linear ODE system having exponential dichotomy and a quasilinear…
We consider the one parameter family $\alpha \mapsto T_\alpha$ ($\alpha \in [0,1)$) of Pomeau-Manneville type interval maps $T_\alpha(x)=x(1+2^\alpha x^\alpha)$ for $x \in [0,1/2)$ and $T_\alpha(x)=2x-1$ for $x \in [1/2, 1]$, with the…
We give two new proofs that the SRB measure of a C^2 path f_t of unimodal piecewise expanding C^3 maps is differentiable at 0 if f_t is tangent to the topological class of f_0. The arguments are more conceptual than the one in our previous…
We study for the first time linear response for random compositions of maps, chosen independently according to a distribution $\PP$. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of…
The average R(t) of a smooth function with respect to the SRB measure of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz. We prove that if f_t is tangent to the topological class of f_0, then…
Arguments inspired by algorithmic information theory predict an inverse relation between the probability and complexity of output patterns in a wide range of input-output maps. This phenomenon is known as \emph{simplicity bias}. By viewing…
When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. Although important in…
We consider a quasi-convex planar domain \Omega with a rectifiable boundary containing an exponential cusp and show that there is no homeomorphism f: \bR^2\to\bR^2 of finite distortion with \exp(\lambda K)\in L_{loc}^{1}(\bR^2) for some…
An element of a group is \emph{reversible} if it is conjugate to its own inverse, and it is \emph{strongly reversible} if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be…
The tent map family is arguably the simplest 1-parametric family of maps with non-trivial dynamics and it is still an active subject of research. In recent works the second author, jointly with J. Yorke, studied the graph and backward…
We consider the dynamics of `nonlinear tent maps': piecewise smooth unimodal maps with nowhere vanishing derivative. We show that if a nonlinear tent map $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently…
We prove ``effective'' linear response for certain classes of non-uniformly expanding random dynamical systems which are not necessarily composed in an i.i.d manner. In applications, the results are obtained for base maps with a sufficient…
On a stack of stable maps, the psi classes are modified by subtracting certain boundary divisors. These modified psi classes are compatible with forgetful morphisms, and are well-suited to enumerative geometry: tangency conditions allow…