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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.

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##### 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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