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Exponential Diophantine approximation and symbolic dynamics

Shigeki Akiyama, Teturo Kamae, Hajime Kaneko

TL;DR

This work develops a general inverse-matrix framework to extend the intertwining between exponential Diophantine approximation and symbolic dynamics beyond Pisot-type bases. By representing ancillary sequences with an infinite-dimensional pair of matrices $\mathcal{A}_f$ and $\mathcal{B}_f$ and introducing the residue-driven sequences $\rho_n^f$, the authors achieve a robust inversion of symbolic encodings for recurrences defined by a polynomial $P(X)$. They establish a set of sharp results for the multiplicative Markoff–Lagrange spectrum-like sets ${\mathcal{L}}^{(k)}(P)$, including closedness for monic hyperbolic $P$, a dichotomy where $0$ is isolated only in the hyperbolic case and density in the non-hyperbolic case, and the existence of intervals near $1/2$ under precise residue and growth conditions. The approach leads to explicit formulas via residues, a framework applicable to polynomials with multiple roots, and yields both discrete and continuous spectral phenomena, with concrete examples and open questions about the spectral geometry and fractal dimensions of these sets.

Abstract

We extend the key formula which intertwines multiplicative Markoff-Lagrange spectrum and symbolic dynamics. The proof uses complex analysis and elucidates the strategy of the problem. Moreover, the new method applies to a wide variety of polynomials possibly having multiple roots. We derive several consequences of this formula, which are expected on the Markoff-Lagrange spectrum.

Exponential Diophantine approximation and symbolic dynamics

TL;DR

This work develops a general inverse-matrix framework to extend the intertwining between exponential Diophantine approximation and symbolic dynamics beyond Pisot-type bases. By representing ancillary sequences with an infinite-dimensional pair of matrices and and introducing the residue-driven sequences , the authors achieve a robust inversion of symbolic encodings for recurrences defined by a polynomial . They establish a set of sharp results for the multiplicative Markoff–Lagrange spectrum-like sets , including closedness for monic hyperbolic , a dichotomy where is isolated only in the hyperbolic case and density in the non-hyperbolic case, and the existence of intervals near under precise residue and growth conditions. The approach leads to explicit formulas via residues, a framework applicable to polynomials with multiple roots, and yields both discrete and continuous spectral phenomena, with concrete examples and open questions about the spectral geometry and fractal dimensions of these sets.

Abstract

We extend the key formula which intertwines multiplicative Markoff-Lagrange spectrum and symbolic dynamics. The proof uses complex analysis and elucidates the strategy of the problem. Moreover, the new method applies to a wide variety of polynomials possibly having multiple roots. We derive several consequences of this formula, which are expected on the Markoff-Lagrange spectrum.
Paper Structure (8 sections, 12 theorems, 147 equations, 1 figure)

This paper contains 8 sections, 12 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic notation
  3. Intertwining formula to symbolic dynamics
  4. Formulae by matrix with the index set ${\mathbb Z}\times {\mathbb Z}$
  5. Calculation of $\rho_n^f$
  6. Applications to Markoff-Lagrange spectrum
  7. Proof of Theorems in Section \ref{['section4']}
  8. Open problems

Key Result

LEMMA 1

Let $(g_1(X),\ldots,g_p(X))\in {\mathbb C}[X]^{r_1}\times {\mathbb C}[X]^{2r_2}$. Assume that $\deg g_j(X)\leq k$ for any $1\leq j\leq p$. (1) if and only if $(g_1(X),\ldots,g_p(X))=(0,\ldots,0)$. (2) if and only if $(g_1(X),\ldots,g_p(X))\in \Xi_k$.

Figures (1)

  • Figure 1: All the mappings commute with the shift.

Theorems & Definitions (26)

  • LEMMA 1
  • proof
  • THEOREM 1
  • LEMMA 2
  • proof
  • LEMMA 3
  • proof
  • LEMMA 4
  • PROPOSITION 1
  • proof
  • ...and 16 more