Invariant sets for substitution

On invariant sets topology

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...

A Homeomorphism Invariant for Substitution

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like ti...

Substitution Invariant Beatty Sequences

with θ irrational and taken to satisfy 0 < θ < 1; plainly this may be assumed without loss of generality. Evidently (fn) is a sequence of zeros and ones. Denote by w0 and w1 words on the alphabet {0, 1} ; that is, finite strings in the letters 0 and 1. Then the sequence (fn) is said to be invariant under the substitution W given by W : 0 −→ w0, 1 −→ w1, if the infinite strings fθ = f1f2f3 . . ....

Substitution invariant cutting sequences

A Homeomorphism Invariant for Substitution Tiling Spaces

Wederive ahomeomorphism invariant for those tiling spaceswhich aremadeby rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in in¢nitely manyorientations.The invariant is a quotient of C4 ech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like tiling s...