Hausdorff Dimension of a Chaotic Set of Shift of a Symbolic Space.

Abstract

For the shift a of the symbolic space ∑ N there exists a subset (called a chaotic set for σ) C of ∑N whose Hausdorff dimension is 1 everywhere (i.e. the Hausdorff dimension of the intersection of C and every non-empty open set of the symbolic space ∑ N is 1), satisfying the condition for any non-empty subset A of the set C, and for any continuous map F: A→∑N there exists a strictly increasing sequence {r n } of positive integers such that the sequence {σ (x)} converges to F(x) for any x∈A. On the other hand, it is shown that in ∑ N every chaotic set for σ has 1-dimensional Hausdorff measure 0.