Štěpán Starosta

Palindromic closures using multiple antimorphisms

A generalized pseudostandard word u, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet A and by a directive sequence of involutory antimorphisms acting on A⁎. Prefixes of u with increasing length are constructed using a pseudopalindromic closure operator. We show that generalized Thue–Morse words tb,m, with b,m∈N and b, m⩾2, are generalized pseudostandard words if and only if tb,m is a periodic word or b⩽m. This extends the result of de Luca and De Luca obtained for the classical Thue–Morse words.

Infinite words rich and almost rich in generalized palindromes

We focus on Θ-rich and almost Θ-rich words over a finite alphabet A, where Θ is an involutive antimorphism over A*. We show that any recurrent almost Θ-rich word u is an image of a recurrent Θ′-rich word under a suitable morphism, where Θ′ is again an involutive antimorphism. Moreover, if the word u is uniformly recurrent, we show that Θ′ can be set to the reversal mapping. We also treat one special case of almost Θ-rich words. We show that every Θ-standard word with seed is an image of an Arnoux-Rauzy word.

Markov constant and quantum instabilities

For a qualitative analysis of spectra of certain two-dimensional rectangular-well quantum systems several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide a new mathematical insight in the phenomenologically relevant occurrence of anomalies in the spectra. Our results may inspire methodical innovations ranging from the description of the stability properties of metamaterials and of certain hiddenly unitary quantum evolution models up to the clarification of the mechanisms of occurrence of ghosts in quantum cosmology.

An algorithm for enumerating all infinite repetitions in a D0L-system

We describe a simple algorithm that finds all primitive words v such that v k is a factor of the language of a given D0L-system for all k. It follows that the number of such words is finite. This polynomial-time algorithm can be also used to decide whether a D0L-system is repetitive.

Morphic images of episturmian words having finite palindromic defect

We study morphisms from certain classes and their action on episturmian words. The first class is P r e t . In general, a morphism of class P r e t can map an infinite word having zero palindromic defect to a word having infinite palindromic defect. We show that the image of an episturmian word, which has zero palindromic defect, under a morphism of class P r e t has always its palindromic defect finite. We also focus on letter-to-letter morphisms to binary alphabet: we show that images of ternary episturmian words under such morphisms have zero palindromic defect. These results contribute to the study of an unsolved question of characterization of morphisms that preserve finite, or even zero, palindromic defect. They also enable us to construct new examples of binary words having zero or finite H -palindromic defect, where H = { Id , R , E , R E } is the group generated by both involutory antimorphisms on a binary alphabet.

Interval Exchange Words and the Question of Hof, Knill, and Simon

We consider words coding non-degenerate 3 interval exchange transformation. It is known that such words contain infinitely many palindromic factors. We show that for any morphism \(\xi \) fixing such a word, either \(\xi \) or \(\xi ^2\) is conjugate to a class P morphism. By this, we provide a new family of palindromic infinite words satisfying the conjecture of Hof, Knill and Simon, as formulated by Tan.

ITINERARIES INDUCED BY EXCHANGE OF THREE INTERVALS

We focus on the exchange T of two intervals with an irrational slope α. For a general subinterval I of the domain of T, the first return time to I takes three values. We describe the structure of the set of return itineraries to I. In particular, we show that it is equal to {R 1 ,R 2 ,R 1 R 2 ,Q} where Q is amicable with R 1 , R 2 or R 1 R 2 .

Fixed points of Sturmian morphisms and their derivated words

Abstract Any infinite uniformly recurrent word u can be written as concatenation of a finite number of return words to a chosen prefix w of u. Ordering of the return words to w in this concatenation is coded by derivated word d u ( w ) . In 1998, Durand proved that a fixed point u of a primitive morphism has only finitely many derivated words d u ( w ) and each derivated word d u ( w ) is fixed by a primitive morphism as well. In our article we focus on Sturmian words fixed by a primitive morphism. We provide an algorithm which to a given Sturmian morphism ψ lists the morphisms fixing the derivated words of the Sturmian word u = ψ ( u ) . We provide a sharp upper bound on length of the list.

On Words with the Zero Palindromic Defect

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word.