January  2007, 18(1): 199-217. doi: 10.3934/dcds.2007.18.199

Random β-expansions with deleted digits

1. 

Department of Mathematics, Utrecht University, Postbus 80.000, 3508 TA Utrecht, Netherlands, Netherlands

Received  June 2006 Revised  November 2006 Published  February 2007

In this paper we define random $\beta$-expansions with digits taken from a given set of real numbers $A= \{ a_1 , \ldots , a_m \}$. We study a generalization of the greedy and lazy expansion and define a function $K$ that generates essentially all $\beta$-expansions with digits belonging to the set $A$. We show that $K$ admits an invariant measure $\nu$ under which $K$ is isomorphic to the uniform Bernoulli shift on $A$.
Citation: Karma Dajani, Charlene Kalle. Random β-expansions with deleted digits. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 199-217. doi: 10.3934/dcds.2007.18.199
[1]

Brian Marcus and Selim Tuncel. Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts. Electronic Research Announcements, 1999, 5: 91-101.

[2]

Christophe Profeta, Frédéric Vrins. Piecewise constant martingales and lazy clocks. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 2-. doi: 10.1186/s41546-019-0036-4

[3]

Jing Qin, Shuang Li, Deanna Needell, Anna Ma, Rachel Grotheer, Chenxi Huang, Natalie Durgin. Stochastic greedy algorithms for multiple measurement vectors. Inverse Problems & Imaging, 2021, 15 (1) : 79-107. doi: 10.3934/ipi.2020066

[4]

John Banks, Thi T. D. Nguyen, Piotr Oprocha, Brett Stanley, Belinda Trotta. Dynamics of spacing shifts. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4207-4232. doi: 10.3934/dcds.2013.33.4207

[5]

John Banks, Piotr Oprocha, Brett Stanley. Transitive sofic spacing shifts. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4743-4764. doi: 10.3934/dcds.2015.35.4743

[6]

Karma Dajani, Cor Kraaikamp, Pierre Liardet. Ergodic properties of signed binary expansions. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 87-119. doi: 10.3934/dcds.2006.15.87

[7]

Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2873-2890. doi: 10.3934/dcds.2020389

[8]

Rainer Buckdahn, Ingo Bulla, Jin Ma. Pathwise Taylor expansions for Itô random fields. Mathematical Control & Related Fields, 2011, 1 (4) : 437-468. doi: 10.3934/mcrf.2011.1.437

[9]

Lu Han, Min Li, Dachuan Xu, Dongmei Zhang. Stochastic-Lazier-Greedy Algorithm for monotone non-submodular maximization. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2607-2614. doi: 10.3934/jimo.2020085

[10]

Philipp Gohlke, Dan Rust, Timo Spindeler. Shifts of finite type and random substitutions. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5085-5103. doi: 10.3934/dcds.2019206

[11]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297

[12]

Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77

[13]

Bing Li, Tuomas Sahlsten, Tony Samuel. Intermediate $\beta$-shifts of finite type. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 323-344. doi: 10.3934/dcds.2016.36.323

[14]

Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451

[15]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[16]

Barbara Kaltenbacher. Periodic solutions and multiharmonic expansions for the Westervelt equation. Evolution Equations & Control Theory, 2021, 10 (2) : 229-247. doi: 10.3934/eect.2020063

[17]

Chaolang Hu, Xiaoming He, Tao Lü. Euler-Maclaurin expansions and approximations of hypersingular integrals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1355-1375. doi: 10.3934/dcdsb.2015.20.1355

[18]

Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593

[19]

Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021006

[20]

Matthew Nicol. Induced maps of hyperbolic Bernoulli systems. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 147-154. doi: 10.3934/dcds.2001.7.147

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]