American Institute of Mathematical Sciences

Advanced
January  2010, 26(1): 333-346. doi: 10.3934/dcds.2010.26.333

Projectional entropy and the electrical wire shift

Michael Schraudner 1,

1. 

Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Piso 7, Santiago de Chile

Received  January 2009 Revised  July 2009 Published  October 2009

In this paper we present an extendible, block gluing $\z^3$ shift of finite type $\w^{el}$ in which the topological entropy equals the $L$-projectional entropy for a two-dimensional sublattice $L$<$\z^3$, even so $\w^{el}$ is not a full $\z$ extension of $\w^{el}_L$. In particular this example shows that Theorem 4.1 of [4] does not generalize to $r$-dimensional sublattices $L$ for $r>1$.
   Nevertheless we are able to reprove and extend the result about one-dimensional sublattices for general $\z^d$ shifts - instead of shifts of finite type - under the same mixing assumption as in [4] and by posing a stronger mixing condition we also obtain the corresponding statement for higher-dimensional sublattices.
Keywords: $\z^d$; multidimensional shift of finite type; projectional entropy; entropy-minimal; non-degenerate..
Mathematics Subject Classification: Primary: 37B50; Secondary: 37B10, 37B4.
Citation: Michael Schraudner. Projectional entropy and the electrical wire shift. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 333-346. doi: 10.3934/dcds.2010.26.333
[1]

Silvère Gangloff. Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 931-988. doi: 10.3934/dcds.2021143
[2]

Kevin McGoff, Ronnie Pavlov. Random $\mathbb{Z}^d$-shifts of finite type. Journal of Modern Dynamics, 2016, 10: 287-330. doi: 10.3934/jmd.2016.10.287
[3]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
[4]

Christopher Hoffman. Subshifts of finite type which have completely positive entropy. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1497-1516. doi: 10.3934/dcds.2011.29.1497
[5]

Piotr Oprocha. Double minimality, entropy and disjointness with all minimal systems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 263-275. doi: 10.3934/dcds.2019011
[6]

Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773
[7]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[8]

Mike Boyle, Sompong Chuysurichay. The mapping class group of a shift of finite type. Journal of Modern Dynamics, 2018, 13: 115-145. doi: 10.3934/jmd.2018014
[9]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
[10]

Ming-Chia Li, Ming-Jiea Lyu. Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 243-252. doi: 10.3934/dcds.2011.30.243
[11]

Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191
[12]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[13]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081
[14]

Fabio Bagagiolo. Optimal control of finite horizon type for a multidimensional delayed switching system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 239-264. doi: 10.3934/dcdsb.2005.5.239
[15]

Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122
[16]

Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
[17]

Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237
[18]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[19]

Masayuki Asaoka, Kenichiro Yamamoto. On the large deviation rates of non-entropy-approachable measures. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4401-4410. doi: 10.3934/dcds.2013.33.4401
[20]

Pieter C. Allaart. An algebraic approach to entropy plateaus in non-integer base expansions. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6507-6522. doi: 10.3934/dcds.2019282

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy