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September  2013, 33(9): 4173-4186. doi: 10.3934/dcds.2013.33.4173

Endomorphisms of Sturmian systems and the discrete chair substitution tiling system

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.
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.

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