Endomorphisms of Sturmian systems and the discrete chair substitution tiling system

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

Abstract
Related Papers

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 and Continuous Dynamical Systems, 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 (1971/72), 129-133.
[2]	Tomasz Downarowicz, The royal couple conceals their mutual relationship: A noncoalescent Toeplitz flow, Israel J. Math., 97 (1997), 239-251. doi: 10.1007/BF02774039.
[3]	Tomasz Downarowicz, Survey of odometers and Toeplitz flows, in "Algebraic and Topological Dynamics," Contemp. Math., 385, Amer. Math. Soc., Providence, RI, (2005), 7-37. doi: 10.1090/conm/385/07188.
[4]	N. Pytheas Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics," eds. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel, Lecture Notes in Mathematics, 1794, Springer-Verlang, Berlin, 2002. doi: 10.1007/b13861.
[5]	Natalie Priebe Frank, A primer of substitution tilings of the Euclidean plane, Expo. Math., 26 (2008), 295-326. doi: 10.1016/j.exmath.2008.02.001.
[6]	Harry Furstenberg, Harvey Keynes and Leonard Shapiro, Prime flows in topological dynamics, Israel J. Math., 14 (1973), 26-38.
[7]	Paul R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2), 43 (1942), 332-350.
[8]	G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375.
[9]	Charles Holton, Charles Radin and Lorenzo Sadun, Conjugacies for tiling dynamical systems, Comm. Math. Phys., 254 (2005), 343-359.
[10]	Douglas Lind and Brian Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995.
[11]	Marston Morse and Gustav A. Hedlund, Symbolic dynamics. II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42.
[12]	James R. Munkres, "Topology: A First Course," Prentice-Hall Inc., Englewood Cliffs, N.J., 1975.
[13]	Jeanette Olli, "Dynamical Systems, Division Point Measures, and Endomorphisms," Ph.D thesis, University of North Carolina at Chapel Hill, 2009.
[14]	Michael E. Paul, Construction of almost automorphic symbolic minimal flows, General Topology and Appl., 6 (1976), 45-56.
[15]	Karl Petersen, On a series of cosecants related to a problem in ergodic theory, Compositio Math., 26 (1973), 313-317.
[16]	Karl Petersen and Leonard Shapiro, Induced flows, Trans. Amer. Math. Soc., 177 (1973), 375-390.
[17]	Charles Radin, Space tilings and substitutions, Geom. Dedicata, 55 (1995), 257-264.
[18]	Charles Radin, Symmetry of tilings of the plane, Bull. Amer. Math. Soc. (N.S.), 29 (1993), 213-217.
[19]	E. Arthur Robinson, Jr., On the table and the chair, Indag. Math. (N.S.), 10 (1999), 581-599.
[20]	E. Arthur Robinson, Jr., Symbolic dynamics and tilings of $\mathbbmathbb{R}^{d}$, in "Symbolic Dynamics and its Applications," Amer. Math. Soc., (2004), 81-119.
[21]	L. Sadun, Tilings, tiling spaces and topology, Philosophical Magazine, 86 (2006), 875-881.
[22]	William A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205-242.

show all references

References:

[1]	Ethan M. Coven, Endomorphisms of substitution minimal sets, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 20 (1971/72), 129-133.
[2]	Tomasz Downarowicz, The royal couple conceals their mutual relationship: A noncoalescent Toeplitz flow, Israel J. Math., 97 (1997), 239-251. doi: 10.1007/BF02774039.
[3]	Tomasz Downarowicz, Survey of odometers and Toeplitz flows, in "Algebraic and Topological Dynamics," Contemp. Math., 385, Amer. Math. Soc., Providence, RI, (2005), 7-37. doi: 10.1090/conm/385/07188.
[4]	N. Pytheas Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics," eds. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel, Lecture Notes in Mathematics, 1794, Springer-Verlang, Berlin, 2002. doi: 10.1007/b13861.
[5]	Natalie Priebe Frank, A primer of substitution tilings of the Euclidean plane, Expo. Math., 26 (2008), 295-326. doi: 10.1016/j.exmath.2008.02.001.
[6]	Harry Furstenberg, Harvey Keynes and Leonard Shapiro, Prime flows in topological dynamics, Israel J. Math., 14 (1973), 26-38.
[7]	Paul R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2), 43 (1942), 332-350.
[8]	G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375.
[9]	Charles Holton, Charles Radin and Lorenzo Sadun, Conjugacies for tiling dynamical systems, Comm. Math. Phys., 254 (2005), 343-359.
[10]	Douglas Lind and Brian Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995.
[11]	Marston Morse and Gustav A. Hedlund, Symbolic dynamics. II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42.
[12]	James R. Munkres, "Topology: A First Course," Prentice-Hall Inc., Englewood Cliffs, N.J., 1975.
[13]	Jeanette Olli, "Dynamical Systems, Division Point Measures, and Endomorphisms," Ph.D thesis, University of North Carolina at Chapel Hill, 2009.
[14]	Michael E. Paul, Construction of almost automorphic symbolic minimal flows, General Topology and Appl., 6 (1976), 45-56.
[15]	Karl Petersen, On a series of cosecants related to a problem in ergodic theory, Compositio Math., 26 (1973), 313-317.
[16]	Karl Petersen and Leonard Shapiro, Induced flows, Trans. Amer. Math. Soc., 177 (1973), 375-390.
[17]	Charles Radin, Space tilings and substitutions, Geom. Dedicata, 55 (1995), 257-264.
[18]	Charles Radin, Symmetry of tilings of the plane, Bull. Amer. Math. Soc. (N.S.), 29 (1993), 213-217.
[19]	E. Arthur Robinson, Jr., On the table and the chair, Indag. Math. (N.S.), 10 (1999), 581-599.
[20]	E. Arthur Robinson, Jr., Symbolic dynamics and tilings of $\mathbbmathbb{R}^{d}$, in "Symbolic Dynamics and its Applications," Amer. Math. Soc., (2004), 81-119.
[21]	L. Sadun, Tilings, tiling spaces and topology, Philosophical Magazine, 86 (2006), 875-881.
[22]	William A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205-242.

[1]	David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[8]	André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems - S, 2009, 2 (2) : 315-324. doi: 10.3934/dcdss.2009.2.315
[10]	Scott Schmieding, Rodrigo Treviño. Random substitution tilings and deviation phenomena. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3869-3902. doi: 10.3934/dcds.2021020
[11]	S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361
[12]	Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229
[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]	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
[15]	Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323
[16]	Benjamin Hellouin de Menibus, Hugo Maturana Cornejo. Necessary conditions for tiling finitely generated amenable groups. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2335-2346. doi: 10.3934/dcds.2020116
[17]	Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485
[18]	Palle Jorgensen, Feng Tian. Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2307-2348. doi: 10.3934/dcdss.2019146
[19]	Xuan Kien Phung. Shadowing for families of endomorphisms of generalized group shifts. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 285-299. doi: 10.3934/dcds.2021116
[20]	Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems and Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Jeanette Olli
on Google Scholar
Jeanette Olli

[Back to Top]