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Pascal Hubert - One of the best experts on this subject based on the ideXlab platform.

  • Rigidity of square-tiled interval exchange transformations
    Journal of Modern Dynamics, 2019
    Co-Authors: Sébastien Ferenczi, Pascal Hubert
    Abstract:

    We look at interval exchange transformations defined as first return maps on the Set of diagonals of a flow of direction \begin{document}$ \theta $\end{document} on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if \begin{document}$ \tan\theta $\end{document} has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and \begin{document}$ \tan\theta $\end{document} has bounded partial quotients, the square-tiled interval exchange transformation \begin{document}$ T $\end{document} is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an Uncountable Set of rigid directional flows and an Uncountable Set of rigid interval exchange transformations.

  • Rigidity of square-tiled interval exchange transformations
    arXiv: Dynamical Systems, 2017
    Co-Authors: Sébastien Ferenczi, Pascal Hubert
    Abstract:

    We look at interval exchange transformations defined as first return maps on the Set of diagonals of a flow of direction $\theta$ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan $\theta$ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan $\theta$ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an Uncountable Set of rigid directional flows and an Uncountable Set of rigid interval exchange transformations.

  • Rigidity of square-tiled interval exchange transformations
    2017
    Co-Authors: Sébastien Ferenczi, Pascal Hubert
    Abstract:

    We look at interval exchange transformations defined as first return maps on the Set of diagonals of a flow of direction θ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan θ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan θ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an Uncountable Set of rigid directional flows and an Uncountable Set of rigid interval exchange transformations.

Sébastien Ferenczi - One of the best experts on this subject based on the ideXlab platform.

  • Rigidity of square-tiled interval exchange transformations
    Journal of Modern Dynamics, 2019
    Co-Authors: Sébastien Ferenczi, Pascal Hubert
    Abstract:

    We look at interval exchange transformations defined as first return maps on the Set of diagonals of a flow of direction \begin{document}$ \theta $\end{document} on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if \begin{document}$ \tan\theta $\end{document} has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and \begin{document}$ \tan\theta $\end{document} has bounded partial quotients, the square-tiled interval exchange transformation \begin{document}$ T $\end{document} is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an Uncountable Set of rigid directional flows and an Uncountable Set of rigid interval exchange transformations.

  • Rigidity of square-tiled interval exchange transformations
    arXiv: Dynamical Systems, 2017
    Co-Authors: Sébastien Ferenczi, Pascal Hubert
    Abstract:

    We look at interval exchange transformations defined as first return maps on the Set of diagonals of a flow of direction $\theta$ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan $\theta$ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan $\theta$ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an Uncountable Set of rigid directional flows and an Uncountable Set of rigid interval exchange transformations.

  • Rigidity of square-tiled interval exchange transformations
    2017
    Co-Authors: Sébastien Ferenczi, Pascal Hubert
    Abstract:

    We look at interval exchange transformations defined as first return maps on the Set of diagonals of a flow of direction θ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan θ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan θ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an Uncountable Set of rigid directional flows and an Uncountable Set of rigid interval exchange transformations.

James Mc Laughlin - One of the best experts on this subject based on the ideXlab platform.

  • The convergence behavior of q-continued fractions on the unit circle
    The Ramanujan Journal, 2006
    Co-Authors: Douglas Bowman, James Mc Laughlin
    Abstract:

    In a previous paper, we showed the existence of an Uncountable Set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value.

  • The convergence behavior of q-continued fractions on the unit circle
    The Ramanujan Journal, 2006
    Co-Authors: Douglas Bowman, James Mc Laughlin
    Abstract:

    In a previous paper, we showed the existence of an Uncountable Set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of q -continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q -continued fraction, G ( q ), in this class, that there is an Uncountable Set of points, Y _ G , on the unit circle such that if y ∊ Y _ G then G ( y ) does not converge to a finite value. We discuss the implications of our theorems for the convergence of other q -continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.

  • On the divergence of the Rogers-Ramanujan continued fraction on the unit circle
    Transactions of the American Mathematical Society, 2003
    Co-Authors: Douglas Bowman, James Mc Laughlin
    Abstract:

    This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0,1) be denoted by [0, e 1 (t), e 2 (t),...] and let the i-th convergent of this continued fraction expansion be denoted by c i (t)/d i (t). Let S = {t ∈ (0, 1): e i+1 (t) > Φ d i (t) infinitely often}, where Φ = (√5 + 1)/2. Let Y S = {exp(2πit): t E S}. It is shown that if y E Y S , then the Rogers-Ramanujan continued fraction R(y) diverges at y. S is an Uncountable Set of measure zero. It is also shown that there is an Uncountable Set of points G C Y S such that if y ∈ G, then R(y) does not converge generally. It is further shown that R(y) does not converge generally for |y| > 1. However we show that R(y) does converge generally if y is a primitive 5m-th root of unity, for some m E N. Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

  • On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle
    arXiv: Number Theory, 2001
    Co-Authors: James Mc Laughlin, Doug Bowman
    Abstract:

    Let the continued fraction expansion of any irrational number $t \in (0,1)$ be denoted by $[0,a_{1}(t),a_{2}(t),...]$ and let the i-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \[ S=\{t \in (0,1): a_{i+1}(t) \geq \phi^{d_{i}(t)} \text{infinitely often}\}, \] where $\phi = (\sqrt{5}+1)/2$. Let $Y_{S} =\{\exp(2 \pi i t): t \in S \}$. It is shown that if $y \in Y_{S}$ then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an Uncountable Set of measure zero. It is also shown that there is an Uncountable Set of points, $G \subSet Y_{S}$, such that if $y \in G$, then R(y) does not converge generally. It is further shown that R(y) does not converge generally for |y| > 1 and that R(y) does converge generally if y is a primitive 5m-th root of unity, some $m \in \mathbb{N}$.

U. A. Rozikov - One of the best experts on this subject based on the ideXlab platform.

  • Nontranslation invariant Gibbs measures for models with Uncountable Set of spin values on a Cayley tree
    Reports on Mathematical Physics, 2018
    Co-Authors: U. A. Rozikov, G. I. Botirov
    Abstract:

    We consider models with nearest-neighbour interactions and with the Set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. It is known that the “splitting Gibbs measures” of the model can be described by solutions of a nonlinear integral equation. Recently, by solving this integral equation some periodic (in particular translation invariant) splitting Gibbs measures were found. In this paper we give three constructions of new Sets of nontranslation invariant splitting Gibbs measures. Our constructions are based on known solutions of the integral equation (1.5) .

  • Four competing interactions for models with an Uncountable Set of spin values on a Cayley tree
    Theoretical and Mathematical Physics, 2017
    Co-Authors: U. A. Rozikov, F. H. Haydarov
    Abstract:

    We consider models with four competing interactions (external field, nearest neighbor, second neighbor, and three neighbors) and an Uncountable Set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique.

  • Non-translation-invariant Gibbs Measures for Models With Uncountable Set of Spin Values on a Cayley Tree
    arXiv: Mathematical Physics, 2017
    Co-Authors: U. A. Rozikov, G. I. Botirov
    Abstract:

    We consider models with nearest-neighbor interactions and with the Set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. Recently, solving this integral equation some periodic (in particular translation-invariant) splitting Gibbs measures were found. In this paper we give three constructions of new Sets of non-translation-invariant splitting Gibbs measures. Our constructions are based on known solutions of the integral equation.

  • Four Competing interactions for models with Uncountable Set of spin values on a Cayley Tree
    arXiv: Mathematical Physics, 2015
    Co-Authors: U. A. Rozikov, F. H. Haydarov
    Abstract:

    In this paper we consider four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with Uncountable (i.e. $[0,1]$) Set of spin values on the Cayley tree of order two. We reduce the problem of describing the "splitting Gibbs measures" of the model to the analysis of solutions to some nonlinear integral equation and study some particular cases for Ising and Potts models. Also we show that periodic Gibbs measures for given models are either translation-invariant or periodic with period two and we give examples of the non-uniqueness of translation-invariant Gibbs measures.

  • Periodic Gibbs measures for models with Uncountable Set of spin values on a Cayley tree
    Infinite Dimensional Analysis Quantum Probability and Related Topics, 2015
    Co-Authors: U. A. Rozikov, F. H. Haydarov
    Abstract:

    We consider models with nearest-neighbor interactions and with the Set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with Uncountable Set of spin values under which the model has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures.

G. I. Botirov - One of the best experts on this subject based on the ideXlab platform.