[Paper Review] Realizability of integer sequences as differences of fixed point count sequences

Natascha Neumaerker|ArXiv.org|May 8, 2009

Mathematical Dynamics and Fractals10 references5 citations

TL;DR

This paper establishes a realizability criterion for integer sequences as differences of fixed point count sequences from a dynamical system and its topological factor. Using Möbius inversion on orbit counts, it proves that an integer sequence is relatively realizable if and only if its orbit transform yields a sequence of integers, extending the classical realizability condition to relative dynamics.

ABSTRACT

A sequence of non-negative integers is exactly realizable as the fixed point counts sequence of a dynamical system if and only if it gives rise to a sequence of non-negative orbit counts. This provides a simple realizability criterion based on the transformation between fixed point and orbit counts. Here, we extend the concept of exact realizability to realizability of integer sequences as differences of the two fixed point counts sequences originating from a dynamical system and a topological factor. A criterion analogous to the one for exact realizability is given and the structure of the resulting set of integer sequences is outlined.

Motivation & Objective

• To extend the concept of exact realizability of integer sequences to relative realizability as differences of fixed point counts between a system and its factor.
• To identify a necessary and sufficient condition for an integer sequence to be realizable as the difference of fixed point counts from a dynamical system and a topological factor.
• To characterize the algebraic and arithmetic structure of the set of relatively realizable sequences using Möbius inversion.
• To demonstrate the criterion on concrete examples from symbolic dynamics, substitution tilings, and S-integer dynamical systems.

Proposed method

• Define relative realizability as the difference between fixed point count sequences of a system and its topological factor.
• Use the Möbius inversion formula to relate fixed point counts to orbit counts: $ c_n = \frac{1}{n} \sum_{d|n} \mu(\frac{n}{d}) a_d $.
• Apply the linear operator $\mathrm{orb}$ to the difference sequence $h = f - g$, yielding $\mathrm{orb}(h) = \mathrm{orb}(f) - \mathrm{orb}(g)$.
• Establish that $h$ is relatively realizable if and only if $\mathrm{orb}(h)$ is a sequence of integers.
• Leverage the inverse relationship between $\mathrm{fix}$ and $\mathrm{orb}$ to reconstruct the orbit count sequence from the difference sequence.
• Illustrate the criterion using examples from symbolic dynamics (e.g., sofic shifts), substitution tilings (Fibonacci, Penrose), and S-integer systems.

Experimental results

Research questions

• RQ1 What conditions must an integer sequence satisfy to be realizable as the difference of fixed point counts from a dynamical system and one of its topological factors?
• RQ2 How does the orbit count sequence of the difference sequence relate to the orbit counts of the original system and its factor?
• RQ3 Can the realizability criterion for exact sequences be generalized to relative differences in the context of dynamical systems?
• RQ4 What structural properties do the sets of relatively realizable sequences exhibit, particularly in terms of arithmetic and algebraic constraints?
• RQ5 How do concrete examples—such as the Fibonacci chain or Penrose tiling with torus parametrization—illustrate the realizability criterion in practice?

Key findings

• An integer sequence $h$ is relatively realizable if and only if $\mathrm{orb}(h)$ is a sequence of integers, generalizing the classical realizability criterion.
• The difference sequence $h = f - g$ between fixed point counts of a system and its factor is realizable precisely when the orbit transform of $h$ yields integer values.
• For the Fibonacci chain, the relative fixed point count sequence is $h_n = (-1)^n$, with corresponding orbit counts $\mathrm{orb}(h) = (-1, 1, 0, 0, \ldots)$.
• In the Penrose tiling example, the relative fixed point count sequence is $(-1, 9, -16, 29, -51, 84, -141, \ldots)$, reflecting non-invertible factor maps.
• For $S$-integer systems with $P$ all primes and $Q = \emptyset$, the relative fixed point sequence is constant $1$, while for $Q = P \setminus \{3\}$, it is $ (0, -2, 0, -2, 0, -8, \ldots) $, with explicit formula $\beta(n) = \frac{1}{2n} \sum_{d|n,\, d\text{ odd}} \mu(d) \cdot 2^{n/d} - \delta_{n,1}$.
• The criterion allows systematic classification of relative fixed point sequences in symbolic and algebraic dynamical systems, particularly where factor maps are not injective.