横截光滑单峰映射族的SRB测度的Whitney-Holder连续性

Modulus of continuity of averages of SRB measures for a transversal family of piecewise expanding unimodal maps

Let $f_t:[0,1] \to [0,1]$ be a family of piecewise expanding unimodal maps with a common critical point that is dense for almost all $t \in [a,b]$. If $μ_t$ is the corresponding SRB measure for $f_t$, we study the regularity of $Γ(t)=\int ϕdμ_t$ when assuming that the family is transversal to the topological classes of these maps, more precisely, we prove that if $J_t(c)=\sum_{k=0}^{\infty} \frac{v_t(f_t^k(c))}{Df_t^k(f_t(c))} \neq 0$ for all $t$, where $v_t(x)=\partial_s f_s(x)|_{s=t}$, then $Γ(t)$ is not Lipschitz for almost all $t\in [a,b]$. Furthermore, we give the exact modulus of continu…

全纯类二次Collet-Eckmann映射族的SRB测度的解析性

Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps

本文证明了：如果f_t是全纯类二次映射族，且所有周期轨道相斥，使得对每个实t，f_t是实Collet-Eckmann S-单峰映射，则记m_t为f_t的唯一绝对连续不变概率测度，映射t -> \int g dm_t对任何实解析函数g都是实解析的.

Central limit theorem for the modulus of continuity of averages of observables on transversal families of piecewise expanding unimodal maps

We prove that the Newton quotient of the average R(t) of a lipschitzian function (with non vanishing variation) with respect to the SRB measure on a transversal family f_t of piecewise expanding unimodal maps, after an appropriated normalization, converges in distribution to the normal distribution.

分段扩张单峰映射线性响应的另一种证明

Alternative proofs of linear response for piecewise expanding unimodal maps

本文给出了单峰分段扩张C^3映射的C^2路ft的SRB测度在0处可微的两个新的证明，如果ft与f0的拓扑类相切.的参数是更多的概念比一个在我们以前的文件，但需要证明持有人连续性的无穷小共轭（一个新的结果，独立的利益），并使用空间的有界p-变化。第一个新的证明给出了高阶可微性的条件是：f_t是足够光滑的，且保持在f_0的拓扑类中，且观测值是足够光滑的（一个新的结果）。此外，这个证明不需要任何关于将SRB度量分解为正则项和奇异项的信息，这使得它可能适合扩展到更高的维度。第二个新的证明允许我们恢复线性响应公式（即，在0处的导数的公式），并给出了关于该公式的附加信息。

Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps

Let $f : [0, 1] \to [0, 1]$ be a piecewise expanding unimodal map of class $C^{k+1}$, with $k \geq 1$, and $μ= ρdx$ the (unique) SRB measure associated to it. We study the regularity of $ρ$. In particular, points $\mathcal{N}$ where $ρ$ is not differentiable has zero Hausdorff dimension, but is uncountable if the critical orbit of $f$ is dense. This improves on a work of Szewc (1984). We also obtain results about higher orders of differentiability of $ρ$ in the sense of Whitney.

分段展开单峰映射的线性响应公式

Linear response formula for piecewise expanding unimodal maps

一个光滑函数关于光滑单参数分段扩张区间映射族f_t的SRB测度的平均R（t）不一定是Lipschitz的.本文证明了：若f_t与f_0的拓扑类相切，则R（t）在零处可微，且其导数与Ruelle猜想所给出的（先验发散）级数的第一作者所提出的重算一致.