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

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…

Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps

We consider C^2 families t->f_t of C^4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure m_t of f_t, as a function of t on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set D of CE parameters, and, for each s in D, a subset D0 of D of polynomially recurrent parameters containing s as a Lebesgue density point, and constants C>1, G >4, so that, for every 1/2-Holder function A (of 1/2-Holder norm |A|) and all t in D0, |\int A dm_t -\int A dm_s| < C |A| |t-s|^{1…

Generalized beta-transformations and the entropy of unimodal maps

Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation $x \mapsto βx ~\pmod 1$, where $β>1$, and replacing some of the branches with branches of constant negative slope. If the orbit of 1 is finite, then the map is Markov, and we call beta (which must be an algebraic number) a generalized Parry number. We show that the Galois conjugates of such beta have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. We give a characterization of the closure o…

Positive Transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps

In this paper we will develop a general approach which shows that generalized "critical relations" of families of locally defined holomorphic maps on the complex plane unfold transversally. The main idea is to define a transfer operator, which is a local analogue of the Thurston pullback operator, using holomorphic motions. Assuming a so-called lifting property is satisfied, we obtain information about the spectrum of this transfer operator and thus about transversality. An important new feature of our method is that it is not global: the maps we consider are only required to be defined and ho…

关于参数连续性模的KAM定理

KAM theorem on modulus of continuity about parameter

本文研究参数仅具连续性的哈密顿系统，在频率映射满足横截与弱凸条件下证明不变环面频率保持性，推导任意Hölder参数扰动的KAM定理并拓展至无穷维，为首项突破Hölder型参数限制的相关研究。

Law of iterated logarithm and invariance principle for one-parameter families of interval maps

We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.