Topological Mixing of Random Substitutions

We investigate topological mixing of compatible random substitutions. For primitive random substitutions on two letters whose second eigenvalue is greater than one in modulus, we identify a simple, computable criterion which is equivalent to topological mixing of the associated subshift. This generalises previous results on deterministic substitutions. In the case of recognisable, irreducible Pisot random substitutions, we show that the associated subshift is not topologically mixing. Without recognisability, we rely on more specialised methods for excluding mixing and we apply these methods to show that the random Fibonacci substitution subshift is not topologically mixing.

[1]  Douglas Lind,et al.  An Introduction to Symbolic Dynamics and Coding , 1995 .

[2]  Philipp Gohlke,et al.  Inflation word entropy for semi-compatible random substitutions , 2019, 1901.07044.

[3]  M. Baake,et al.  Binary Constant-Length Substitutions and Mahler Measures of Borwein Polynomials , 2017, 1711.02492.

[4]  Timo Spindeler,et al.  Ergodic frequency measures for random substitutions , 2018, 1810.11122.

[5]  Gwendolyn S. Tadeo,et al.  Zeckendorf representations and mixing properties of sequences , 2019, Tsukuba Journal of Mathematics.

[6]  Jean-Marc Luck,et al.  Quasiperiodicity and randomness in tilings of the plane , 1989 .

[7]  Tony Samuel,et al.  Measure theoretic entropy of random substitution subshifts , 2021 .

[8]  Valérie Berthé,et al.  On the Pisot Substitution Conjecture , 2015 .

[9]  Boris Adamczewski,et al.  Balances for fixed points of primitive substitutions , 2003, Theor. Comput. Sci..

[10]  R. Kenyon,et al.  Topological mixing for substitutions on two letters , 2004, Ergodic Theory and Dynamical Systems.

[11]  F. Dekking,et al.  Mixing properties of substitutions , 1978 .

[12]  Val'erie Berth'e,et al.  Beyond substitutive dynamical systems: S-adic expansions , 2013, 1309.3960.

[13]  Jean-Marie Dumont,et al.  Systemes de Numeration et Fonctions Fractales Relatifs aux Substitutions , 1989, Theor. Comput. Sci..

[14]  F. M. Dekking,et al.  The spectrum of dynamical systems arising from substitutions of constant length , 1978 .

[15]  Brigitte Mossé,et al.  Puissances de mots et reconnaissabilité des point fixes d'une substitution , 1992, Theor. Comput. Sci..

[16]  Dan Rust,et al.  Dynamical systems arising from random substitutions , 2017, Indagationes Mathematicae.

[17]  Michael Baake,et al.  What is Aperiodic Order , 2002, 1512.05104.

[18]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[19]  Lorenzo Sadun,et al.  Topology of tiling spaces , 2008 .

[20]  Valérie Berthé,et al.  Balancedness and coboundaries in symbolic systems , 2018, Theor. Comput. Sci..

[21]  Timo Spindeler,et al.  Shifts of finite type and random substitutions , 2017, Discrete & Continuous Dynamical Systems - A.

[22]  Dan Rust,et al.  Periodic points in random substitution subshifts , 2018, Monatshefte für Mathematik.

[23]  H. Poincaré,et al.  Les méthodes nouvelles de la mécanique céleste , 1899 .

[24]  Edmonton,et al.  Diffraction of compatible random substitutions in one dimension , 2017, Indagationes Mathematicae.