29 research outputs found
A Note on Symmetries in the Rauzy Graph and Factor Frequencies
We focus on infinite words with languages closed under reversal. If
frequencies of all factors are well defined, we show that the number of
different frequencies of factors of length n+1 does not exceed 2C(n+1)-2C(n)+1.Comment: 7 page
Factor frequencies in languages invariant under more symmetries
The number of frequencies of factors of length in a recurrent aperiodic
infinite word does not exceed 3\Delta \C(n), where \Delta \C (n) is the
first difference of factor complexity, as shown by Boshernitzan. Pelantov\'a
together with the author derived a better upper bound for infinite words whose
language is closed under reversal. In this paper, we further diminish the upper
bound for uniformly recurrent infinite words whose language is invariant under
all elements of a finite group of symmetries and we prove the optimality of the
obtained upper bound.Comment: 13 page
Palindromic richness for languages invariant under more symmetries
For a given finite group consisting of morphisms and antimorphisms of a
free monoid , we study infinite words with language closed under
the group . We focus on the notion of -richness which describes words
rich in generalized palindromic factors, i.e., in factors satisfying
for some antimorphism . We give several
equivalent descriptions which are generalizations of know characterizations of
rich words (in the terms of classical palindromes) and show two examples of
-rich words
A Note On -Rauzy Graphs for the Infinite Fibonacci Word
The -Rauzy graph of order for any infinite word is a directed graph
in which an arc is formed if the concatenation of the word
and the suffix of of length is a subword of the infinite word.
In this paper, we consider one of the important aperiodic recurrent words, the
infinite Fibonacci word for discussion. We prove a few basic properties of the
-Rauzy graph of the infinite Fibonacci word. We also prove that the
-Rauzy graphs for the infinite Fibonacci word are strongly connected.Comment: 10 pages, 4 figure
ITINERARIES INDUCED BY EXCHANGE OF THREE INTERVALS
We focus on a generalization of the three gap theorem well known in the framework of exchange of two intervals. For the case of three intervals, our main result provides an analogue of this result implying that there are at most 5 gaps. To derive this result, we give a detailed description of the return times to a subinterval and the corresponding itineraries