اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدComputation of Topological Entropy via $phi$-expansion, an Inverse Problem for the Dynamical Systems $beta x+alpha mod 1$
162
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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 exposition 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.
قيم البحث
اقرأ أيضاً
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 $beta$.
Let $mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,{f_n}_{n=1}^{+infty})$
vanishes, then so does that of its induced system $(mathcal{M}(X),{f_n}_{n=1}^{+infty})$; moreover, once the topological entropy of $(X,{f_n}_{n=1}^{+infty})$ is positive, that of its induced system $(mathcal{M}(X),{f_n}_{n=1}^{+infty})$ jumps to infinity. In contrast to Bowens inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension.
We survey an area of recent development, relating dynamics to theoretical computer science. We discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems. We will focus on central objects of the theory
of dynamics, as invariant measures and invariant sets, showing that even if they can be computed with arbitrary precision in many interesting cases, there exists some cases in which they can not. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem).
We introduce the notion of Bohr chaoticity, which is a topological invariant for topological dynamical systems, and which is opposite to the property required by Sarnaks conjecture. We prove the Bohr chaoticity for all systems which have a horseshoe
and for all toral affine dynamical systems of positive entropy, some of which dont have a horseshoe. But uniquely ergodic dynamical systems are not Bohr chaotic.
In this paper, we investigate the embeddings for topological flows. We prove an embedding theorem for discrete topological system. Our results apply to suspension flows via constant function, and for this case we show an embedding theorem for suspens
ion flows and give a new proof of Gutman-Jin embedding theorem.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
