Critical Connectedness of Thin Arithmetical Discrete Planes

The critical thickness of an arithmetical discrete plane refers to the infimum thickness that preserves its 2-connectedness. This infimum thickness can be computed thanks to a multidimensional continued fraction algorithm, namely the fully subtractive algorithm. We provide a characterization of the discrete planes with critical thickness that contain the origin and that are 2-connected.

[1]  Pierre Arnoux,et al.  The Rauzy Gasket , 2013 .

[2]  Jean-Pierre Reveillès,et al.  Géométrie discrète, calcul en nombres entiers et algorithmique , 1991 .

[3]  Zoltán Fülöp,et al.  Developments in Language Theory , 2003, Lecture Notes in Computer Science.

[4]  Eric Andres.  Discrete linear objects in dimension n: the standard model , 2003, Graph. Model..

[5]  Eric Andres,et al.  Discrete Analytical Hyperplanes , 1997, CVGIP Graph. Model. Image Process..

[6]  Xavier Provençal,et al.  Facet Connectedness of Discrete Hyperplanes with Zero Intercept: The General Case , 2014, DGCI.

[7]  Jean-Luc Toutant,et al.  Minimal arithmetic thickness connecting discrete planes , 2009, Discret. Appl. Math..

[8]  Valentin E. Brimkov,et al.  Connectivity of discrete planes , 2004, Theor. Comput. Sci..

[9]  Fritz Schweiger,et al.  Multidimensional continued fractions , 2000 .

[10]  Shunji Ito,et al.  Discrete planes, Z2-actions, Jacobi-Perron algorithm and substitutions , 2006 .

[11]  Tomasz Nowicki,et al.  Infinite clusters and critical values in two-dimensional circle percolation , 1989 .

[12]  Cor Kraaikamp,et al.  Ergodic properties of a dynamical system arising from percolation theory , 1995, Ergodic Theory and Dynamical Systems.

[13]  Robbert Fokkink,et al.  On Schweiger’s problems on fully subtractive algorithms , 2011 .

[14]  Valérie Berthé,et al.  Substitutive Arnoux-Rauzy sequences have pure discrete spectrum , 2011, ArXiv.

[15]  Jean-Luc Toutant,et al.  On the Connecting Thickness of Arithmetical Discrete Planes , 2009, DGCI.

[16]  Pierre Arnoux,et al.  Two-dimensional iterated morphisms and discrete planes , 2004, Theor. Comput. Sci..

[17]  M. Mišík,et al.  Oxford University Press , 1968, PMLA/Publications of the Modern Language Association of America.

[18]  Arnaldo Nogueira,et al.  Multidimensional Continued Fractions. By Fritz Schweiger. Oxford Science Publications , 2002, Ergodic Theory and Dynamical Systems.

[19]  Jean-Luc Toutant,et al.  On the Connectedness of Rational Arithmetic Discrete Hyperplanes , 2006, DGCI.

[20]  Luca Q. Zamboni,et al.  Fine and Wilf words for any periods , 2003 .

[21]  Thomas Fernique,et al.  Bidimensional Sturmian Sequences and Substitutions , 2005, Developments in Language Theory.

[22]  Ronald W. J. Meester.  An algorithm for calculating critical probabilities and percolation functions in percolation models defined by rotations , 1989 .

[23]  Laurent Vuillon,et al.  Geometric Palindromic Closure , 2012 .

[24]  P. Arnoux,et al.  Pisot substitutions and Rauzy fractals , 2001 .

[25]  Makoto Ohtsuki,et al.  Modified Jacobi-Perron Algorithm and Generating Markov Partitions for Special Hyperbolic Toral Automorphisms , 1993 .