bibkey: grosjeanhuillet2016carleman
authors: Nicolas Grosjean; Thierry Huillet
year: 2016
title: "Some combinatorial aspects of discrete non-linear population dynamics"
doi: 10.1016/j.chaos.2016.10.004
url: "https://arxiv.org/abs/1610.05919v1"
claim: "For an iterated power series map with the Carleman matrix P(k,k')=[x^{k'}]φ(x)^k, the critical case α₁=1 is treated by a normalized Jordan-chain matrix W with PW=W(I+S), and the paper conjectures that W⁻¹(k,k')=σ(k−1,k'), a sum over strictly increasing index paths of products of entries of P."
strata_touched:
  - D5/S3/StatisticalMechanics/CarlemanJordanChainInverse
license: citation-only
triage: anchor

Grosjean and Huillet, Some combinatorial aspects of discrete non-linear population dynamics

N. Grosjean and T. Huillet, Chaos, Solitons & Fractals 93, 71–79 (2016); arXiv:1610.05919v1. Quotations are from the v1 TeX source.

For $\varphi (x)={\sum }_{k\ge 1}{\alpha }_k{x}^k$ with ${\alpha }_1\ne 0$, the paper uses the upper-triangular Carleman matrix $P(k,k^{\prime })=[x^{k^{\prime }}]\varphi (x{)}^k$, $k^{\prime }\ge k\ge 1$, to linearize the iteration $x_{n+1}=\varphi (x_n)$. In the critical case ${\alpha }_1=1$ the matrix $P$ has unit diagonal and is not diagonalizable; the paper constructs columns ${\mathbf{w}}_k$ with $P{\mathbf{w}}_1={\mathbf{w}}_1$, $P{\mathbf{w}}_k={\mathbf{w}}_{k-1}+{\mathbf{w}}_k$ for $k>1$, normalized by ${\mathbf{w}}_1=(1,0,\dots )$, ${\mathbf{w}}_k(k^{\prime })=0$ for $k^{\prime }>k$ and ${\mathbf{w}}_k(1)=0$ for $k>1$, through a recursion that divides by $P(k^{\prime }-1,k^{\prime })=(k^{\prime }-1){\alpha }_2$. With $W$ the matrix of these columns, $\sigma (0,k^{\prime })={\delta }_{1,k^{\prime }}$, $\sigma (k,k^{\prime })=0$ for $k^{\prime }\le k$, and $\sigma (k,k^{\prime })={\sum }_{1=i_1<\cdots <i_{k+1}=k^{\prime }}{\prod }_{l=1}^{k}P(i_l,i_{l+1})$ for $k^{\prime }>k\ge 1$, the paper states:

Then, we conjecture that $W^{-1}(k,k^{\prime })=\sigma (k-1,k^{\prime })$, $k^{\prime }\ge k\ge 1$.

Verified locator

Source: https://arxiv.org/abs/1610.05919v1

• DOI: https://doi.org/10.1016/j.chaos.2016.10.004 (Chaos, Solitons & Fractals 93, 71–79; title, authors, journal, volume and pages from Crossref).
• URL: https://arxiv.org/abs/1610.05919v1. The e-print member snlmatrixRR.tex has 1387 lines and md5 8c08ca10132a50d7c92ce541ba3d2240. Locations:
  • the power series and the Carleman matrix, Eq. (2), l. 126–141;
  • the critical case and the Jordanization, l. 400–437;
  • $\sigma$ and the conjecture, l. 441–452.