[Paper Review] Patterns of Negative Shifts and Beta-Shifts

Sergi Elizalde, Katherine Moore|arXiv (Cornell University)|Dec 14, 2015

Cellular Automata and Applications10 references3 citations

TL;DR

This paper extends the study of permutation patterns in dynamical systems to negative β-shifts, introducing a combinatorial framework to compute the minimal β > 1 such that a given permutation π is realized by the negative β-shift map. It provides a formula for the negative shift-complexity B̄(π) as the largest real root of a specific polynomial P̄π(β), generalizing prior results on positive β-shifts and integer negative shifts to the real negative β case.

ABSTRACT

The $β$-shift is the transformation from the unit interval to itself that maps $x$ to the fractional part of $βx$. Permutations realized by the relative order of the elements in the orbits of these maps have been studied for positive integer values of $β$ and for real values $β>1$. In both cases, a combinatorial description of the smallest positive value of $β$ needed to realize a permutation is provided. In this paper we extend these results to the case of negative $β$, both in the integer and in the real case. Negative $β$-shifts are related to digital expansions with negative real bases, studied by Ito and Sadahiro, and Liao and Steiner.

Motivation & Objective

• To extend the theory of permutation patterns in β-shifts to the case of negative real bases β > 1.
• To determine the smallest β > 1 such that a given permutation π is realized by the negative β-shift map T₋β.
• To generalize prior results on integer negative shifts and positive β-shifts to the full real case for negative β.
• To characterize the set of allowed patterns of the negative β-shift and compute their shift-complexity.
• To establish a connection between negative β-expansions of real numbers and the realization of permutations via orbit orderings.

Proposed method

• Define the negative β-shift map T₋β: (0,1] → (0,1] by T₋β(x) = 1 - {βx}, which generalizes the standard β-shift to negative base β > 1.
• Use the concept of admissible words in the negative β-expansion system to represent real numbers in base -β.
• For a given permutation π, construct a word w in the admissible set W₋β that induces π via orbit ordering.
• Derive a polynomial P̄π(β) whose largest real root β ≥ 1 gives the minimal β such that π is realized by T₋β.
• Apply the theory of alternating order and infinite words to determine the critical value β = B̄(π) as the largest real solution to f_w[ℓ,∞)(x) = 1.
• Use the fact that B̄(π) = 1 if the tail of the word w is lexicographically smaller than the alternating order of the standard word u.

Experimental results

Research questions

• RQ1 What is the minimal β > 1 such that a given permutation π is realized by the negative β-shift map T₋β?
• RQ2 How can the set of allowed patterns of the negative β-shift be characterized combinatorially?
• RQ3 What is the relationship between negative β-expansions of real numbers and the realization of permutations via orbit orderings?
• RQ4 Can the shift-complexity B̄(π) be computed algorithmically for any permutation π?
• RQ5 How does the structure of the polynomial P̄π(β) relate to the shape and structure of the permutation π?

Key findings

• The negative shift-complexity B̄(π) of a permutation π is equal to the largest real root of a polynomial P̄π(β), which is explicitly constructed from the permutation’s structure.
• For permutations of length 4, B̄(π) ranges from 1 to approximately 2.247, with specific values tied to the roots of polynomials like β³ - 2β² + β - 1.
• For permutations of length 5, B̄(π) ranges from 1 to approximately 3.243, with values such as 2.7693 corresponding to the largest real root of β³ - 3β² + β - 1.
• When no real root β ≥ 1 exists for P̄π(β), B̄(π) = 1, which occurs precisely when the tail of the associated word is lexicographically smaller than the alternating order of the standard word.
• The construction of the word w that induces π ensures that B̄(π) is the minimal β > 1 such that π ∈ Allow(T₋β), and this value is computable via root-finding on P̄π(β).
• The method generalizes prior results on positive β-shifts and integer negative shifts, providing a unified framework for real negative β > 1.