Yuru Zou

Unique expansion of points of a class of self-similar sets with overlaps

For q >1, the set F q of real numbers which can be expanded in base q with respect to the digit set {0,1, q } is just a self-similar set with overlaps. We consider the subset of F q whose elements have a unique expansion and calculate its Hausdorff dimension for the case where .

Self-similar structure on the intersection of middle-(1 − 2β) Cantor sets with β ∈ (1/3,1/2)

Let ?? be the middle-(1 ? 2?) Cantor set with ? (1/3, 1/2). We give all real numbers t with unique {?1, 0, 1}-code such that the intersections ?? ? (??+t) are self-similar sets. For a given ? (1/3, 1/2), a criterion is obtained to check whether or not a t [?1, 1] has the unique {?1, 0, 1}-code from both geometric and algebraic views.

COLORFUL PATTERNS WITH DISCRETE PLANAR SYMMETRIES FROM DYNAMICAL SYSTEMS

Automatic generation of colored patterns with discrete planar symmetries is considered from a dynamical systems point of view. Invariant mappings with such symmetries are constructed to serve as the density functions for the generation of colorful images.

On a problem of countable expansions

Abstract For a real number q ∈ ( 1 , 2 ) and x ∈ [ 0 , 1 / ( q − 1 ) ] , the infinite sequence ( d i ) is called a q-expansion of x if x = ∑ i = 1 ∞ d i q i , d i ∈ { 0 , 1 } for all i ≥ 1 . For m = 1 , 2 , ⋯ or ℵ 0 we denote by B m the set of q ∈ ( 1 , 2 ) such that there exists x ∈ [ 0 , 1 / ( q − 1 ) ] having exactly m different q-expansions. It was shown by Sidorov [18] that q 2 : = min ⁡ B 2 ≈ 1.71064 , and later asked by Baker [1] whether q 2 ∈ B ℵ 0 ? In this paper we provide a negative answer to this question and conclude that B ℵ 0 is not a closed set. In particular, we give a complete description of x ∈ [ 0 , 1 / ( q 2 − 1 ) ] having exactly two different q 2 -expansions.

COLORFUL SYMMETRIC IMAGES IN THREE-DIMENSIONAL SPACE FROM DYNAMICAL SYSTEMS

Functions that are invariant with respect to the tetrahedral and cubic symmetries are determined. These invariant mappings are applied to serve as the density functions for automatic generation of the colorful images with such symmetries in three dimensional space from a dynamical systems point of view.

ON SMALL BASES FOR WHICH 1 HAS COUNTABLY MANY EXPANSIONS

Let

On the cardinality of β-expansions of some numbers

q\in(1,2)

On small bases which admit points with two expansions

. A

Intersecting nonhomogeneous Cantor sets with their translations

q

A class of Sierpinski carpets with overlaps

-expansion of a number