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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 & 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
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