arXiv subjects

Florent Nguema Ndong

Publications and source records attributed to Florent Nguema Ndong.

4 recordsLinked to original sources

arXiv2026

Exchange of Intervals and Intrinsic Ergodicity of the Negative Beta shift

This work highlights a peculiar phenomenon of interval exchange. Considering a real number beta less than -1, the negative beta-shift is coded if and only if its absolute value is greater than the golden ratio. We study an increasing sequence of algebraic integers with limit-1 and the absolute value of the first term equals to the golden ratio such that for a base x taken in the interval of consicutive terms of this sequence, the measure of the maximal entropy is carried by the image of a beta-shift, with the golden ratio les than the absolute value of beta, under the mapping of an injective substitution.

math.DS↗

arXiv2021

Contribution on The Intrinsic Ergodicity of the Negative Beta-shift

Let $ β$ be a real number less than -1. In this paper, we prove the uniqueness of the measure with maximal entropy of the negative $β$-shift. Endowed with the shift, this symbolic dynamical system is coded under certain conditions, but in all cases, it is shown that the measure with maximal entropy is carried by a support coded by a recurrent positive code. One of the difference between the positive and the negative $β$-shift is the existence of gaps in the system for certain negative values of $ β$ . These are intervals of negative $β$-representations (cylinders) negligible with respect to the measure with maximal entropy, which is a measure of Champernown.

math.DS↗

arXiv2018

The $(-β)$-shift and associated Zeta Function

Given a real number $ β> 1$, we study the associated $ (-β)$-shift introduced by S. Ito and T. Sadahiro. We compares some aspects of the $(-β)$-shift to the $β$-shift. When the expansion in base $ -β$ of $ -\fracβ{β+1} $ is periodic with odd period or when $ β$ is strictly less than the golden ratio, the $ (-β)$-shift, as defined by S. Ito and T. Sadahiro cannot be coded because its language is not transitive. This intransitivity of words explains the existence of gaps in the interval. We observe that an intransitive word appears in the $(-β)$-expansion of a real number taken in the gap. Furthermore, we determine the Zeta function $ζ_{-β}$ of the $(-β)$-transformation and the associated lap-counting function $L_{T_{-β}}$. These two functions are related by $ ζ_{-β}=(1-z^2)L_{T_{-β}}$. We observe some similarities with the zeta function of the $β$-transformation. The function $ζ_{-β}$ is meromorphic in the unit disk, is holomorphic in the open disk $ \{z |z| < \frac{1}β \}$, has a simple pole at $ \frac{1}β$ and no other singularities $ z $ such that $\|z| = \frac{1}β$. We also note an influence of gaps ($β$ less than the golden ratio) on the zeta function. In factors of the denominator of $ζ_{-β}$, the coefficients count the words generating gaps.

math.DS↗

arXiv2017

On the Lyndon Dynamical Systems

Given a totally finite ordered alphabet $ A $, endowing the set of words over $ A $ with the alternating lexicographic order, we define a new class of Lyndon words. We study the fundamental properties of the associated symbolic dynamical systems called Lyndon system. We derive some fundamental properties of the beta-shift with negative base by relating it with the Lyndon system. We find, independently of W. Steiner's method, the conditions for which a word is the $(-β)$-expansion of $-\fracβ{β+ 1} $ for some $ β> 1$.

math.DS↗