Discrete Mathematics & Theoretical Computer Science |

- 1 Department of Mathematics [Dartmouth]

For a real number $β >1$, we say that a permutation $π$ of length $n$ is allowed (or realized) by the $β$-shift if there is some $x∈[0,1]$ such that the relative order of the sequence $x,f(x),\ldots,f^n-1(x)$, where $f(x)$ is the fractional part of $βx$, is the same as that of the entries of $π$ . Widely studied from such diverse fields as number theory and automata theory, $β$-shifts are prototypical examples of one-dimensional chaotic dynamical systems. When $β$ is an integer, permutations realized by shifts have been recently characterized. In this paper we generalize some of the results to arbitrary $β$-shifts. We describe a method to compute, for any given permutation $π$ , the smallest $β$ such that $π$ is realized by the $β$-shift.

Source: HAL:hal-01215112v1

Volume: DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

Section: Proceedings

Published on: January 1, 2011

Imported on: January 31, 2017

Keywords: beta-shift,forbidden pattern,consecutive pattern,shift map,real base expansion,dynamical system,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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