اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn Hausdorff dimension of the set of closed orbits for a cylindrical transformation
109
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We deal with Besicovitchs problem of existence of discrete orbits for transitive cylindrical transformations $T_varphi:(x,t)mapsto(x+alpha,t+varphi(x))$ where $Tx=x+alpha$ is an irrational rotation on the circle $T$ and $varphi:TtoR$ is continuous, i.e. we try to estimate how big can be the set $D(alpha,varphi):={xinT:|varphi^{(n)}(x)|to+inftytext{as}|n|to+infty}$. We show that for almost every $alpha$ there exists $varphi$ such that the Hausdorff dimension of $D(alpha,varphi)$ is at least $1/2$. We also provide a Diophantine condition on $alpha$ that guarantees the existence of $varphi$ such that the dimension of $D(alpha,varphi)$ is positive. Finally, for some multidimensional rotations $T$ on $T^d$, $dgeq3$, we construct smooth $varphi$ so that the Hausdorff dimension of $D(alpha,varphi)$ is positive.
قيم البحث
اقرأ أيضاً
We prove that the Hausdorff dimension of the set of three-period orbits in classical billiards is at most one. Moreover, if the set of three-period orbits has Hausdorff dimension one, then it has a tangent line at almost every point.
We compute the Hausdorff dimension of limit sets generated by 3-dimensional self-affine mappings with diagonal matrices of the form A_{ijk}=Diag(a_{ijk}, b_{ij}, c_{i}), where 0<a_{ijk}le b_{ij}le c_i<1. By doing so we show that the variational principle for the dimension holds for this class.
As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension 3, we introduce a $C^2$-open set of diffeomorphisms of whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fract
al set of Hausdorff dimension nearly $1$. Our proof employs the thicknesses of Cantor sets.
We show that the graph of the classical Weierstrass function $sum_{n=0}^infty lambda^n cos (2pi b^n x)$ has Hausdorff dimension $2+loglambda/log b$, for every integer $bge 2$ and every $lambdain (1/b,1)$. Replacing $cos(2pi x)$ by a general non-const
ant $C^2$ periodic function, we obtain the same result under a further assumption that $lambda b$ is close to $1$.
A function which is transcendental and meromorphic in the plane has at least two singular values. On one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either
$2$ or $1/2$. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in $[0,2]$ (cf. cite{ac1}). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than $4$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
