Topological Entropy Of A Class Of Constrained Systems

In the field of topologically dynamical systems,symbolic dynamical systems is an important topic.The research of subshift of finite type is of great significance in theory and application,and then has attracted much attention.For the subshift of finite type,there have been a lot of research results.In particular,the run-length-limited(d,k)constraint system is applied to a variety of storage systems and industrial complex machine operations.It is defined as a constraint system with two symbols of {0,1},where the number of consecutive“1”and consecutive “0”numbers in the sequence is at least d and at least k.This type of systems has been studied by the scholars.They are a class of space limited subshift,and one can calculate their topological entropies by their characteristic polynomials.In fact,the topological entropies of different constraint systems can be equal,and the equations must be one of the following forms:C(d,2d)= C(d + 1,3d + 1),C(d + 1,?)= C(d,2d + 1),d ? 0,C(1,2)= C(2,4)= C(3,7)= C(4,?).In this paper,we consider a class of constrained systems named double upper bounds(p,q)-constrained systems((p,q)-DUB systems in brief),which are a class of subshifts of finite type.The terminology of(p,q)constraint means continuous codes in sequences with up to p zeros and q ones.Denote C(p,q)by the topological entropy of a(p,q)-DUB systems.The value of the topological entropy of the(d,k)constrained system can be obtained by the root of the corresponding polynomial,but it is difficult to compare the sizes of the entropies between different constraint systems directly by the numerical sizes of d and k.However,for(p,q)–DUB,We can find an effective method to easily compare the topological entropies of any two or more(p,q)constrained systems.In fact,we can sort all topological entropies of(p,q)constrained systems.The conclusion is:0 = C(1,1)< C(1,2)<...< C(1,?)= C(2,2)< C(2,3)<...< C(2,?)= C(3,3)< C(3,4)<......< C(?,?)= ln 2.In particular,C(p,?)= C(p + 1,p + 1)are the only possible equalities among the topological entropies of(p,q)-DUB systems.