Dynamical properties of the Pascal adic transformation

We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a two-symbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing.

[1]  S. Kakutani,et al.  Invariant measures and orbits of dissipative transformations , 1972 .

[2]  B. Pitskel,et al.  Poisson limit law for Markov chains , 1991, Ergodic Theory and Dynamical Systems.

[3]  F. Paccaut.  Statistics of return times for weighted maps of the interval , 2000 .

[4]  Richard C. Bollinger,et al.  Lucas's theorem and some related results for extended pascal triangles , 1990 .

[5]  M. Queffélec.  Substitution dynamical systems, spectral analysis , 1987 .

[6]  Sébastien Ferenczi,et al.  Rank and symbolic complexity , 1996, Ergodic Theory and Dynamical Systems.

[7]  Sébastien Ferenczi,et al.  Complexity of sequences and dynamical systems , 1999, Discret. Math..

[8]  A. N. Livshits.  A sufficient condition for weak mixing of substitutions and stationary adic transformations , 1988 .

[9]  Valentin Afraimovich,et al.  Local dimensions for Poincaré recurrences , 2000 .

[10]  Boris Solomyak,et al.  On the spectral theory of adic transformations , 1992 .

[11]  伊藤 俊次.  A construction of transversal flows for maximal Markov automorphisms = マルコフ変換に対する横断的流れの横成の研究 , 1979 .

[12]  INVARIANT COCYCLES, RANDOM TILINGS AND THE SUPER-K AND STRONG MARKOV PROPERTIES , 1997 .

[13]  É. Lucas.  Theorie des Fonctions Numeriques Simplement Periodiques , 1878 .

[14]  J. Cassaigne,et al.  Complexity and growth for polygonal billiards , 2001, math/0109208.

[15]  K. Petersen.  Disjointness and weak mixing of minimal sets , 1970 .

[16]  Xavier Méla.  A class of nonstationary adic transformations , 2006 .

[17]  Christian F. Skau,et al.  Substitutional dynamical systems, Bratteli diagrams and dimension groups , 1999, Ergodic Theory and Dynamical Systems.

[18]  A. Forrest.  K-groups associated with substitution minimal systems , 1997 .

[19]  Timo Jolivet,et al.  Substitutions , 2002, Peach State.

[20]  Z. Coelho.  ASYMPTOTIC LAWS FOR SYMBOLIC DYNAMICAL SYSTEMS , 2005 .

[21]  P. Halmos.  Lectures on ergodic theory , 1956 .

[22]  Shunji Ito.  A Construction of Transversal Flows for Maximal Markov Automorphisms , 1978 .

[23]  Anatoly M. Vershik,et al.  Adic models of ergodic transformations, spectral theory, substitutions, and related topics , 1992 .

[24]  On fluctuations and the exponential statistics of return times , 2001 .

[25]  Y. Lacroix.  Possible limit laws for entrance times of an ergodic aperiodic dynamical system , 2002 .

[26]  A. Maass,et al.  Limit laws of entrance times for low-complexity Cantor minimal systems , 2001, 0808.0602.

[27]  K. Schmidt,et al.  Symmetric Gibbs measures , 1996, math/9604240.

[28]  A. Vershik,et al.  A theorem on the Markov periodic approximation in ergodic theory , 1985 .

[29]  S. Ferenczi.  Systems of finite rank , 1997 .

[30]  A. Vershik,et al.  Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group , 1985 .

[31]  Eigenvalue theorems in topological transformation groups , 1969 .

[32]  E. Janvresse,et al.  The Pascal adic transformation is loosely Bernoulli , 2004 .

[33]  I. Putnam,et al.  Ordered Bratteli diagrams, dimension groups and topological dynamics , 1992 .

[34]  Anatoly M. Vershik,et al.  Asymptotic theory of characters of the symmetric group , 1981 .

[35]  Karl Petersen,et al.  Information Compression and Retention in Dynamical Processes , 2002 .

[36]  P. Collet,et al.  ASYMPTOTIC DISTRIBUTION OF ENTRANCE TIMES FOR EXPANDING MAPS OF THE INTERVAL , 1995 .

[37]  Tail-fields of products of random variables and ergodic equivalence relations , 1999, Ergodic Theory and Dynamical Systems.

[38]  François Blanchard,et al.  SUBSTITUTION SUBSHIFTS AND BRATTELI DIAGRAMS , 2000 .