American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory
  • DCDS Home
  • This Issue
  • Next Article
    Simple skew category algebras associated with minimal partially defined dynamical systems
September  2013, 33(9): 4173-4186. doi: 10.3934/dcds.2013.33.4173

Endomorphisms of Sturmian systems and the discrete chair substitution tiling system

Jeanette Olli 1,

1. 

Dominican University, 7900 W. Division Street, River Forest, IL 60305, United States

Received  July 2010 Revised  February 2013 Published  March 2013

When looking at a dynamical system, a natural question to ask is what are its endomorphisms. Using Coven's work in [1] on the endomorphisms of dynamical systems generated by substitutions of equal length on {0,1} as a guide, we fully describe the endomorphisms for a class of almost automorphic symbolic dynamical systems provided there are certain conditions on the set where the factor map fails to be 1-1. While this result does have conditions on both the dynamical system and the factor map, it applies to Sturmian systems and generalized Sturmian systems. We also prove a similar result for a particular 2-dimensional system with a $\mathbb{Z}^2$-action, the discrete chair substitution tiling system.
Keywords: Endomorphisms, Sturmian, tiling., chair substitution.
Mathematics Subject Classification: Primary: 37B05, 37B50, 37B10; Secondary: 54H2.
Citation: Jeanette Olli. Endomorphisms of Sturmian systems and the discrete chair substitution tiling system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4173-4186. doi: 10.3934/dcds.2013.33.4173
References:
[1]

Ethan M. Coven, Endomorphisms of substitution minimal sets,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 20 (): 129.   Google Scholar

[2]

Tomasz Downarowicz, The royal couple conceals their mutual relationship: A noncoalescent Toeplitz flow,, Israel J. Math., 97 (1997), 239.  doi: 10.1007/BF02774039.  Google Scholar

[3]

Tomasz Downarowicz, Survey of odometers and Toeplitz flows,, in, 385 (2005), 7.  doi: 10.1090/conm/385/07188.  Google Scholar

[4]

N. Pytheas Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics,", eds. V. Berthé, 1794 (2002).  doi: 10.1007/b13861.  Google Scholar

[5]

Natalie Priebe Frank, A primer of substitution tilings of the Euclidean plane,, Expo. Math., 26 (2008), 295.  doi: 10.1016/j.exmath.2008.02.001.  Google Scholar

[6]

Harry Furstenberg, Harvey Keynes and Leonard Shapiro, Prime flows in topological dynamics,, Israel J. Math., 14 (1973), 26.   Google Scholar

[7]

Paul R. Halmos and J. von Neumann, Operator methods in classical mechanics. II,, Ann. of Math. (2), 43 (1942), 332.   Google Scholar

[8]

G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system,, Math. Systems Theory, 3 (1969), 320.   Google Scholar

[9]

Charles Holton, Charles Radin and Lorenzo Sadun, Conjugacies for tiling dynamical systems,, Comm. Math. Phys., 254 (2005), 343.   Google Scholar

[10]

Douglas Lind and Brian Marcus, "An Introduction to Symbolic Dynamics and Coding,", Cambridge University Press, (1995).   Google Scholar

[11]

Marston Morse and Gustav A. Hedlund, Symbolic dynamics. II. Sturmian trajectories,, Amer. J. Math., 62 (1940), 1.   Google Scholar

[12]

James R. Munkres, "Topology: A First Course,", Prentice-Hall Inc., (1975).   Google Scholar

[13]

Jeanette Olli, "Dynamical Systems, Division Point Measures, and Endomorphisms,", Ph.D thesis, (2009).   Google Scholar

[14]

Michael E. Paul, Construction of almost automorphic symbolic minimal flows,, General Topology and Appl., 6 (1976), 45.   Google Scholar

[15]

Karl Petersen, On a series of cosecants related to a problem in ergodic theory,, Compositio Math., 26 (1973), 313.   Google Scholar

[16]

Karl Petersen and Leonard Shapiro, Induced flows,, Trans. Amer. Math. Soc., 177 (1973), 375.   Google Scholar

[17]

Charles Radin, Space tilings and substitutions,, Geom. Dedicata, 55 (1995), 257.   Google Scholar

[18]

Charles Radin, Symmetry of tilings of the plane,, Bull. Amer. Math. Soc. (N.S.), 29 (1993), 213.   Google Scholar

[19]

E. Arthur Robinson, Jr., On the table and the chair,, Indag. Math. (N.S.), 10 (1999), 581.   Google Scholar

[20]

E. Arthur Robinson, Jr., Symbolic dynamics and tilings of $\mathbbR^d$,, in, (2004), 81.   Google Scholar

[21]

L. Sadun, Tilings, tiling spaces and topology,, Philosophical Magazine, 86 (2006), 875.   Google Scholar

[22]

William A. Veech, Point-distal flows,, Amer. J. Math., 92 (1970), 205.   Google Scholar

show all references

References:
[1]

Ethan M. Coven, Endomorphisms of substitution minimal sets,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 20 (): 129.   Google Scholar

[2]

Tomasz Downarowicz, The royal couple conceals their mutual relationship: A noncoalescent Toeplitz flow,, Israel J. Math., 97 (1997), 239.  doi: 10.1007/BF02774039.  Google Scholar

[3]

Tomasz Downarowicz, Survey of odometers and Toeplitz flows,, in, 385 (2005), 7.  doi: 10.1090/conm/385/07188.  Google Scholar

[4]

N. Pytheas Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics,", eds. V. Berthé, 1794 (2002).  doi: 10.1007/b13861.  Google Scholar

[5]

Natalie Priebe Frank, A primer of substitution tilings of the Euclidean plane,, Expo. Math., 26 (2008), 295.  doi: 10.1016/j.exmath.2008.02.001.  Google Scholar

[6]

Harry Furstenberg, Harvey Keynes and Leonard Shapiro, Prime flows in topological dynamics,, Israel J. Math., 14 (1973), 26.   Google Scholar

[7]

Paul R. Halmos and J. von Neumann, Operator methods in classical mechanics. II,, Ann. of Math. (2), 43 (1942), 332.   Google Scholar

[8]

G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system,, Math. Systems Theory, 3 (1969), 320.   Google Scholar

[9]

Charles Holton, Charles Radin and Lorenzo Sadun, Conjugacies for tiling dynamical systems,, Comm. Math. Phys., 254 (2005), 343.   Google Scholar

[10]

Douglas Lind and Brian Marcus, "An Introduction to Symbolic Dynamics and Coding,", Cambridge University Press, (1995).   Google Scholar

[11]

Marston Morse and Gustav A. Hedlund, Symbolic dynamics. II. Sturmian trajectories,, Amer. J. Math., 62 (1940), 1.   Google Scholar

[12]

James R. Munkres, "Topology: A First Course,", Prentice-Hall Inc., (1975).   Google Scholar

[13]

Jeanette Olli, "Dynamical Systems, Division Point Measures, and Endomorphisms,", Ph.D thesis, (2009).   Google Scholar

[14]

Michael E. Paul, Construction of almost automorphic symbolic minimal flows,, General Topology and Appl., 6 (1976), 45.   Google Scholar

[15]

Karl Petersen, On a series of cosecants related to a problem in ergodic theory,, Compositio Math., 26 (1973), 313.   Google Scholar

[16]

Karl Petersen and Leonard Shapiro, Induced flows,, Trans. Amer. Math. Soc., 177 (1973), 375.   Google Scholar

[17]

Charles Radin, Space tilings and substitutions,, Geom. Dedicata, 55 (1995), 257.   Google Scholar

[18]

Charles Radin, Symmetry of tilings of the plane,, Bull. Amer. Math. Soc. (N.S.), 29 (1993), 213.   Google Scholar

[19]

E. Arthur Robinson, Jr., On the table and the chair,, Indag. Math. (N.S.), 10 (1999), 581.   Google Scholar

[20]

E. Arthur Robinson, Jr., Symbolic dynamics and tilings of $\mathbbR^d$,, in, (2004), 81.   Google Scholar

[21]

L. Sadun, Tilings, tiling spaces and topology,, Philosophical Magazine, 86 (2006), 875.   Google Scholar

[22]

William A. Veech, Point-distal flows,, Amer. J. Math., 92 (1970), 205.   Google Scholar

[1]

David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287
[2]

Rui Pacheco, Helder Vilarinho. Statistical stability for multi-substitution tiling spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4579-4594. doi: 10.3934/dcds.2013.33.4579
[3]

Marcy Barge, Sonja Štimac, R. F. Williams. Pure discrete spectrum in substitution tiling spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 579-597. doi: 10.3934/dcds.2013.33.579
[4]

Marcy Barge. Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1159-1173. doi: 10.3934/dcds.2016.36.1159
[5]

Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483
[6]

N. Romero, A. Rovella, F. Vilamajó. Dynamics of vertical delay endomorphisms. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 409-422. doi: 10.3934/dcdsb.2003.3.409
[7]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[8]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351
[9]

Rafael De La Llave, A. Windsor. An application of topological multiple recurrence to tiling. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 315-324. doi: 10.3934/dcdss.2009.2.315
[10]

S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361
[11]

Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229
[12]

Joshua P. Bowman, Slade Sanderson. Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces. Journal of Modern Dynamics, 2020, 16: 109-153. doi: 10.3934/jmd.2020005
[13]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684
[14]

Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323
[15]

Benjamin Hellouin de Menibus, Hugo Maturana Cornejo. Necessary conditions for tiling finitely generated amenable groups. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2335-2346. doi: 10.3934/dcds.2020116
[16]

Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485
[17]

Palle Jorgensen, Feng Tian. Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2307-2348. doi: 10.3934/dcdss.2019146
[18]

Jeong-Yup Lee, Boris Solomyak. On substitution tilings and Delone sets without finite local complexity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3149-3177. doi: 10.3934/dcds.2019130
[19]

Brett M. Werner. An example of Kakutani equivalent and strong orbit equivalent substitution systems that are not conjugate. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 239-249. doi: 10.3934/dcdss.2009.2.239
[20]

Junxiang Li, Yan Gao, Tao Dai, Chunming Ye, Qiang Su, Jiazhen Huo. Substitution secant/finite difference method to large sparse minimax problems. Journal of Industrial & Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences