The Sizes Of Multiply Xiong Chaotic Sets And Mean Li-Yorke Chaotic Sets Along A Sequence

We consider the "size" of the multiply Xiong chaotic set for full shift over finite symbols and the Gauss system.We also consider the "size" of two kinds of chaotic sets including the multiply Xiong chaotic set and the mean Li-Yorke chaotic set along polynomial sequences for any ?-transformations.We will use the Hausdorff dimension and the topological entropy to describe the "size"of chaotic sets.Let(X,d)be a metric space,f:X? X be a continuous self-map.A subset E of X containing at least two points is said to be multiply Xiong chaotic if for any d ?N,any subset A of E,and continuous functions gj:A?X for j=1,2,…,d,there exists an increasing sequence {qk}k=1? of positive integers such that for every x ? A,(?)fj·qk(x)=gj(x),j=1,2,…,d.The first main result is that we construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere(i.e.,the intersection of this multiply Xiong chaotic set and any arbitrary open set has full Hausdorff dimension)for the full shift over finite symbols.We also show that this kind of chaotic set can be contained in the multiply proximal cell of each point.Then,by the same method,we obtain a similar result for the full shift over countable symbols.Applying the above-mentioned result in the Gauss system,we prove that under the Gauss map,for every irrational number in the interval[0,1),its multiply proximal cell contains a multiply Xiong chaotic set with full Haus-dorff dimension everywhere.Then we prove that there exists a multiply Xiong chaotic set with full topological entropy everywhere(i.e.,the intersection of this multiply Xiong chaotic set and any ar-bitrary open set has full topological entropy)for any ?-transformation.In order to prove this result,We show that for positively expansive systems with specification property,there exists a multiply Xiong chaotic set with full topological entropy everywhere.?-transformation is not continuous and it does not satisfy specification property for all?>1.So,we construct a multiply Xiong chaotic set in ?-shifts by adopting the same method as in the construction for positively expansive systems with specification prop-erty.To finish the proof,we map it to the interval[0,1)and keep its topological entropy being In ? at the meanwhile.Let {an}n=1? be a sequence of non-negative integers.Another kind of chaotic set that we considered is called a mean Li-Yorke chaotic set along the sequence {an} An uncountable subset C is said to be a mean Li-Yorke chaotic set along the sequence {an},if both(?) and(?)hold for any two distinct points x and y in C.The last main result is that for any ?>1,we construct a mean Li-Yorke chaotic set along polynomial sequences(the degree of this polynomial is not less than three)with full Hausdorff dimension and full topological entropy for the ?-transformation.First,construct a mean Li-Yorke chaotic set along the polynomial sequence for ?-shifts and then maps it to the interval[0,1)by keeping the Hausdorff dimension being 1 at the meanwhile.During the construction of a mean Li-Yorke chaotic set along the polynomial sequence for ?-shifts,several good properties of admissible words in ?-shifts are used frequently.