Noble Means in Symbolic Dynamics

• Noble means are a family of inflation multipliers and substitution rules derived from quadratic equations, generalizing Fibonacci and golden mean sequences.
• Random noble means substitutions generate both deterministic and stochastic symbolic dynamics with explicitly computable ergodic, spectral, and entropy properties.
• Applications include modeling aperiodic order and mixed spectral phenomena through explicit Delone and Meyer set constructions in mathematical physics.

Noble, in the context of mathematical physics and symbolic dynamics, refers to a special class of inflation multipliers, substitution rules, and the associated random dynamical systems characterized by so-called "noble means." This terminology arises from the algebraic and combinatorial properties of certain infinite words, tilings, and their generalizations, notably extending the well-known Fibonacci chain. Noble means inflation and its randomizations yield deterministically and stochastically generated hulls with explicitly computable ergodic, spectral, and entropy properties, central to the study of aperiodic order and mixed spectral phenomena (Baake et al., 2012).

1. Noble Mean Inflation Multipliers and Substitution Systems

For each integer $m \geq 1$, the noble mean $\lambda_m$ is defined as the Perron–Frobenius root of the quadratic equation: $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$ with algebraic conjugate $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$. The number $\lambda_m$ is a purely periodic continued fraction, $[m; m, m, m, \dots]$. This algebraic setting unifies classical cases: for $m=1$, $\lambda_1 = \frac{1+\sqrt{5}}{2}$ recovers the golden mean (Fibonacci inflation).

Let $A = \{a, b\}$; for each $0 \leq i \leq m$, define the primitive substitution $\lambda_m$0 by: $\lambda_m$1 All such substitutions share the $\lambda_m$2 matrix $\lambda_m$3, with eigenvalues $\lambda_m$4 and $\lambda_m$5. The principal (right) eigenvector gives letter or tile frequencies.

2. Construction and Properties of Random Noble Means Substitutions

A random local mixture is specified by a probability vector $\lambda_m$6 ($\lambda_m$7, $\lambda_m$8), defining the random noble means substitution (RNMS) $\lambda_m$9 through

• $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$0 with probability $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$1 use $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$2, for $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$3
• $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$4 deterministically

Each application at any position is made independently across symbols, yielding a stochastic process on $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$5 and, via suitable limiting procedures, on bi-infinite sequences. Thus, the associated state space (symbolic hull) $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$6 comprises all bi-infinite sequences whose every finite subword appears with positive probability in some realization of iterated RNMS.

3. Dynamical Systems Structure and Invariant Measures

The topological dynamical system $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$7 is defined by the closure, under the left shift $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$8, of the set of RNMS-generated sequences. For each $$x^2 - m\,x - 1 = 0 \qquad\Longrightarrow\qquad \lambda_m = \frac{m + \sqrt{m^2+4}}{2}$$9, the system admits a block-induced substitution $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$0 on the set $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$1 of legal words of length $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$2. The corresponding primitive substitution matrix $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$3 has Perron–Frobenius eigenvalue $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$4 and strictly positive eigenvector $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$5.

An ergodic, shift-invariant measure $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$6 is constructed via: $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$7 for the cylinder set $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$8 of words with $\lambda_m' = \frac{m - \sqrt{m^2+4}}{2}$9 in positions $\lambda_m$0. The system $\lambda_m$1 is ergodic, with weak mixing along suitable subsequences by the strong law of large numbers.

4. Dynamical Invariants: Entropy and Frequency Measures

Topological entropy $\lambda_m$2 quantifies the exponential growth rate of distinct exact realizations of the substitution applied to $\lambda_m$3: if $\lambda_m$4 denotes these words with common length $\lambda_m$5 (where $\lambda_m$6), then

$\lambda_m$7

For $\lambda_m$8 (the random Fibonacci case) this specializes to $\lambda_m$9.

The measure $[m; m, m, m, \dots]$0 defined above is the unique ergodic invariant probability under shift, ensuring almost-sure frequencies of all patches match the stationary distribution, and the system is metric-entropy maximizing.

5. Diffraction Spectrum and Meyer Set Structure

Associating real line tilings to symbolic sequences by assigning tile lengths $[m; m, m, m, \dots]$1 for $[m; m, m, m, \dots]$2 and $[m; m, m, m, \dots]$3 for $[m; m, m, m, \dots]$4, the left-endpoints generate a Delone set $[m; m, m, m, \dots]$5, often described via a cut-and-project/Meyer set construction. The autocorrelation measure $[m; m, m, m, \dots]$6 of $[m; m, m, m, \dots]$7 exists almost surely, and its Fourier transform $[m; m, m, m, \dots]$8 decomposes as

$[m; m, m, m, \dots]$9

where

$m=1$0

Here $m=1$1, and the amplitudes $m=1$2 are averages over the tile type positions. In the random case, there is a non-trivial absolutely continuous component: in the $m=1$3 case (random Fibonacci), this part is

$m=1$4

where $m=1$5 is an explicit Radon–Nikodym density computable via the variance of the exponential sums

$m=1$6

with recursions for $m=1$7, $m=1$8.

6. Special Cases: Random Fibonacci and Associated Formulas

For $m=1$9, the substitutions are

$\lambda_1 = \frac{1+\sqrt{5}}{2}$0

with inflation matrix $\lambda_1 = \frac{1+\sqrt{5}}{2}$1 and $\lambda_1 = \frac{1+\sqrt{5}}{2}$2. The RNMS picks $\lambda_1 = \frac{1+\sqrt{5}}{2}$3 with probability $\lambda_1 = \frac{1+\sqrt{5}}{2}$4 and $\lambda_1 = \frac{1+\sqrt{5}}{2}$5 with $\lambda_1 = \frac{1+\sqrt{5}}{2}$6. Explicit results for $\lambda_1 = \frac{1+\sqrt{5}}{2}$7, $\lambda_1 = \frac{1+\sqrt{5}}{2}$8, $\lambda_1 = \frac{1+\sqrt{5}}{2}$9, and $A = \{a, b\}$0 follow, in particular a recursion for the amplitude vector: $A = \{a, b\}$1 which converges as $A = \{a, b\}$2.

7. Significance, Mixed Spectra, and Applications

The random noble means family provides the first fully explicit construction of a locally randomized, but globally self-similar, substitution-based dynamical system where the pure-point (quasiperiodic) part of the spectrum survives, yet positive topological entropy and an absolutely continuous diffraction component arise from independent local substitutions. The structure of $A = \{a, b\}$3-inflation and Meyer-set geometry is preserved, while the addition of local randomness introduces controlled disorder and spectral mixing. This supplies exactly solvable models for mixed spectral phenomena in aperiodic order, supporting exploration of the interplay between algebraic order, entropy, and spectral types in symbolic and tiling dynamical systems (Baake et al., 2012).