اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA Point is Normal for Almost All Maps $beta x + alpha mod 1$ or Generalized $beta$-Maps
195
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the map $T_{alpha,beta}(x):= beta x + alpha mod 1$, which admits a unique probability measure of maximal entropy $mu_{alpha,beta}$. For $x in [0,1]$, we show that the orbit of $x$ is $mu_{alpha,beta}$-normal for almost all $(alpha,beta)in[0,1)times(1,infty)$ (Lebesgue measure). Nevertheless we construct analytic curves in $[0,1)times(1,infty)$ along them the orbit of $x=0$ is at most at one point $mu_{alpha,beta}$-normal. These curves are disjoint and they fill the set $[0,1)times(1,infty)$. We also study the generalized $beta$-maps (in particular the tent map). We show that the critical orbit $x=1$ is normal with respect to the measure of maximal entropy for almost all $beta$.
قيم البحث
اقرأ أيضاً
We give an algorithm, based on the $phi$-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is the solve an inverse problem for the dynamical systems $beta x+alpha mod1$.The first part is an expositio
n of the $phi$-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the $phi$-expansion, necessary and sufficient conditions, which are different from those in Parrys paper.
We consider the minimal average action (Mathers $beta$ function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the $beta$-function
associated to a standard-like twist map admits a unique $C^1$-holomorphic complex extension, which coincides with this function on the set of real diophantine frequencies.
Poincares last geometric theorem (Poincare-Birkhoff Theorem) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essen
tial simple closed curve intersects its image under $f$ at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincares geometric theorem, our result also has some applications to reversible systems.
For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^ntoR$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation pro
perties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.
We describe an algorithm for distinguishing hyperbolic components in the parameter space of quadratic rational maps with a periodic critical point. We then illustrate computer images of the hyperbolic components of the parameter spaces V1 - V4, which
were produced using our algorithm. We also resolve the singularities of the projective closure of V5 by blowups, giving an alternative proof that as an algebraic curve, the geometric genus of V5 is 1. This explains why we are unable to produce an image for V5.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
