17 research outputs found
Morphisms, Symbolic sequences, and their Standard Forms
Morphisms are homomorphisms under the concatenation operation of the set of
words over a finite set. Changing the elements of the finite set does not
essentially change the morphism. We propose a way to select a unique
representing member out of all these morphisms. This has applications to the
classification of the shift dynamical systems generated by morphisms. In a
similar way, we propose the selection of a representing sequence out of the
class of symbolic sequences over an alphabet of fixed cardinality. Both methods
are useful for the storing of symbolic sequences in databases, like The On-Line
Encyclopedia of Integer Sequences. We illustrate our proposals with the
-symbol Fibonacci sequences
Intersections of homogeneous Cantor sets and beta-expansions
Let be the -part homogeneous Cantor set with
. Any string with
such that is called a code of . Let
be the set of having a unique code,
and let be the set of which make the intersection a
self-similar set. We characterize the set in a
geometrical and algebraical way, and give a sufficient and necessary condition
for . Using techniques from beta-expansions, we
show that there is a critical point , which is a
transcendental number, such that has positive
Hausdorff dimension if , and contains countably
infinite many elements if . Moreover, there exists a
second critical point
such that
has positive Hausdorff dimension if
, and contains countably infinite many elements if
.Comment: 23 pages, 4 figure
The structure of base phi expansions
In the base phi expansion any natural number is written uniquely as a sum of
powers of the golden mean with coefficients 0 and 1, where it is required that
the product of two consecutive digits is always 0. We tackle the problem of
describing how these expansions look like. We classify the positive parts of
the base phi expansions according to their suffices, and the negative parts
according to their prefixes, specifying the sequences of occurrences of these
digit blocks. Here the situation is much more complex than for the Zeckendorf
expansions, where any natural number is written uniquely as a sum of Fibonacci
numbers with coefficients 0 and 1, where, again, it is required that the
product of two consecutive digits is always 0. In a previous work we have
classified the Zeckendorf expansions according to their suffices. It turned out
that if we consider the suffices as labels on the Fibonacci tree, then the
numbers with a given suffix in their Zeckendorf expansion appear as generalized
Beatty sequences in a natural way on this tree. We prove that the positive
parts of the base phi expansions are a subsequence of the sequence of
Zeckendorf expansions, giving an explicit formula in terms of a generalized
Beatty sequence. The negative parts of the base phi expansions no longer appear
lexicographically. We prove that all allowed digit blocks appear, and determine
the order in which they do appear
The structure of base phi expansions
In the base phi expansion, a natural number is written uniquely as a sum of powers of the golden mean with coefficients 0 and 1, where it is required that the product of two consecutive digits is always 0. We tackle the problem of describing these expansions in detail. We classify the positive parts of the base phi expansions according to their suffixes, and the negative parts according to their prefixes, specifying the sequences of occurrences of these digit blocks. We prove that the positive parts of the base phi expansions are a subsequence of the sequence of Zeckendorf expansions, giving an explicit formula in terms of a generalized Beatty sequence. The negative parts of the base phi expansions no longer appear lexicographically. We prove that all allowed digit blocks appear, and determine the order in which they do appear