Directional uniformities, periodic points, and entropy

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December 2015, 20(10): 3525-3545. doi: 10.3934/dcdsb.2015.20.3525

Directional uniformities, periodic points, and entropy

Richard Miles ^1, and Thomas Ward ^2,

1.	Uppsala Universitet, Lägerhyddsvägen 1, Hus 1, 5 och 7, 75106 Uppsala, Sweden
2.	Durham University, Durham DH1 3LE, United Kingdom

Received November 2014 Revised March 2015 Published September 2015

Abstract
Related Papers

Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological $\mathbb{Z}^d$-actions) contain within them structures on many scales, and in particular contain many actions of $\mathbb{Z}^k$ for $1\le k\le d$. Familiar dynamical invariants for homeomorphisms, like entropy and periodic point data, become more complex and permit multiple definitions. We briefly survey some of these and other related invariants in the setting of algebraic $\mathbb{Z}^d$-actions, showing how, even in settings where the natural entropy as a $\mathbb{Z}^d$-action vanishes, a powerful theory of directional entropy and periodic points can be built. An underlying theme is uniformity in dynamical invariants as the direction changes, and the connection between this theory and problems in number theory; we explore this for several invariants. We also highlight Fried's notion of average entropy and its connection to uniformities in growth properties, and prove a new relationship between this entropy and periodic point growth in this setting.

Keywords: directional entropy, expansive subdynamics, algebraic dynamics., Directional dynamics.

Mathematics Subject Classification: Primary: 37B40, 37C25; Secondary: 37P35, 37C4.

Citation: Richard Miles, Thomas Ward. Directional uniformities, periodic points, and entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3525-3545. doi: 10.3934/dcdsb.2015.20.3525

References:

[1]	L. M. Abramov, The entropy of an automorphism of a solenoidal group, Teor. Veroyatnost. i Primenen, 4 (1959), 249-254.
[2]	N. Ailon and Z. Rudnick, Torsion points on curves and common divisors of $a^k-1$ and $b^k-1$, Acta Arith., 113 (2004), 31-38. doi: 10.4064/aa113-1-3.
[3]	A. Baker, Transcendental Number Theory, 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511565977.
[4]	P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle, Acta Arith., 18 (1971), 355-369.
[5]	M. Boyle and D. Lind, Expansive subdynamics, Trans. Amer. Math. Soc., 349 (1997), 55-102. doi: 10.1090/S0002-9947-97-01634-6.
[6]	V. Chothi, G. Everest and T. Ward, $S$-integer dynamical systems: Periodic points, J. Reine Angew. Math., 489 (1997), 99-132. doi: 10.1515/crll.1997.489.99.
[7]	P. M. Cohn, Algebraic Numbers and Algebraic Functions, Chapman and Hall Mathematics Series, Chapman & Hall, London, 1991. doi: 10.1007/978-1-4899-3444-4.
[8]	P. Corvaja and U. Zannier, A lower bound for the height of a rational function at $S$-unit points, Monatsh. Math., 144 (2005), 203-224. doi: 10.1007/s00605-004-0273-0.
[9]	E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith., 34 (1979), 391-401.
[10]	M. Einsiedler and D. Lind, Algebraic $\mathbbZ^d$-actions of entropy rank one, Trans. Amer. Math. Soc., 356 (2004), 1799-1831. doi: 10.1090/S0002-9947-04-03554-8.
[11]	M. Einsiedler, D. Lind, R. Miles and T. Ward, Expansive subdynamics for algebraic $\mathbbZ^d$-actions, Ergodic Theory Dynam. Systems, 21 (2001), 1695-1729. doi: 10.1017/S014338570100181X.
[12]	G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Universitext, Springer-Verlag London, Ltd., London, 1999. doi: 10.1007/978-1-4471-3898-3.
[13]	D. Fried, Entropy for smooth abelian actions, Proc. Amer. Math. Soc., 87 (1983), 111-116. doi: 10.1090/S0002-9939-1983-0677244-7.
[14]	S. Friedland, Entropy of graphs, semigroups and groups}, in Ergodic Theory of $Z^d$ Actions (Warwick, 1993-1994), London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996, 319-343. doi: 10.1017/CBO9780511662812.013.
[15]	W. Geller and M. Pollicott, An entropy for $\mathbb Z^2$-actions with finite entropy generators, Dedicated to the memory of Wiesław Szlenk, Fund. Math., 157 (1998), 209-220.
[16]	A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms, arXiv:1211.0987, to appear.
[17]	A. Katok, S. Katok and F. R. Hertz, The Fried average entropy and slow entropy for actions of higher rank abelian groups, Geometric and Functional Analysis, 24 (2014), 1204-1228. doi: 10.1007/s00039-014-0284-5.
[18]	B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems, 9 (1989), 691-735. doi: 10.1017/S0143385700005290.
[19]	F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A561-A563.
[20]	D. Lind, K. Schmidt and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math., 101 (1990), 593-629. doi: 10.1007/BF01231517.
[21]	D. A. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy, Ergodic Theory Dynam. Systems, 8 (1988), 411-419. doi: 10.1017/S0143385700004545.
[22]	R. Miles, A natural boundary for the dynamical zeta function for commuting group automorphisms, Proc. Amer. Math. Soc., 143 (2015), 2927-2933. doi: 10.1090/S0002-9939-2015-12515-4.
[23]	R. Miles, Zeta functions for elements of entropy rank-one actions, Ergodic Theory Dynam. Systems, 27 (2007), 567-582. doi: 10.1017/S0143385706000794.
[24]	R. Miles, Finitely represented closed-orbit subdynamics for commuting automorphisms, Ergodic Theory Dynam. Systems, 30 (2010), 1787-1802. doi: 10.1017/S0143385709000741.
[25]	R. Miles, Synchronization points and associated dynamical invariants, Trans. Amer. Math. Soc., 365 (2013), 5503-5524. doi: 10.1090/S0002-9947-2013-05829-1.
[26]	R. Miles, M. Staines and T. Ward, Dynamical invariants for group automorphisms, Contemp. Math., 631 (2015), 231-258. doi: 10.1090/conm/631/12606.
[27]	R. Miles and T. Ward, Periodic point data detects subdynamics in entropy rank one, Ergodic Theory Dynam. Systems, 26 (2006), 1913-1930. doi: 10.1017/S014338570600054X.
[28]	R. Miles and T. Ward, Uniform periodic point growth in entropy rank one, Proc. Amer. Math. Soc., 136 (2008), 359-365. doi: 10.1090/S0002-9939-07-09018-1.
[29]	R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc., 137 (2009), 1499-1507. doi: 10.1090/S0002-9939-08-09649-4.
[30]	R. Miles and T. Ward, A dichotomy in orbit growth for commuting automorphisms, J. Lond. Math. Soc. (2), 81 (2010), 715-726. doi: 10.1112/jlms/jdq010.
[31]	R. Miles and T. Ward, A directional uniformity of periodic point distribution and mixing, Discrete Contin. Dyn. Syst., 30 (2011), 1181-1189. doi: 10.3934/dcds.2011.30.1181.
[32]	J. Milnor, On the entropy geometry of cellular automata, Complex Systems, 2 (1988), 357-385.
[33]	G. Morris and T. Ward, Entropy bounds for endomorphisms commuting with $K$ actions, Israel J. Math., 106 (1998), 1-11. doi: 10.1007/BF02773458.
[34]	M. Pollicott, A note on the growth of periodic points for commuting toral automorphisms, ISRN Geometry, 2012 (2012), Article ID 165808, 15 pages. doi: 10.5402/2012/165808.
[35]	K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-0277-2.
[36]	K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei, Invent. Math., 111 (1993), 69-76. doi: 10.1007/BF01231280.
[37]	K. R. Yu, Linear forms in $p$-adic logarithms. II, Compositio Math., 74 (1990), 15-113; Available from: http://www.numdam.org/item?id=CM_1990__74_1_15_0.

show all references

References:

[1]	L. M. Abramov, The entropy of an automorphism of a solenoidal group, Teor. Veroyatnost. i Primenen, 4 (1959), 249-254.
[2]	N. Ailon and Z. Rudnick, Torsion points on curves and common divisors of $a^k-1$ and $b^k-1$, Acta Arith., 113 (2004), 31-38. doi: 10.4064/aa113-1-3.
[3]	A. Baker, Transcendental Number Theory, 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511565977.
[4]	P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle, Acta Arith., 18 (1971), 355-369.
[5]	M. Boyle and D. Lind, Expansive subdynamics, Trans. Amer. Math. Soc., 349 (1997), 55-102. doi: 10.1090/S0002-9947-97-01634-6.
[6]	V. Chothi, G. Everest and T. Ward, $S$-integer dynamical systems: Periodic points, J. Reine Angew. Math., 489 (1997), 99-132. doi: 10.1515/crll.1997.489.99.
[7]	P. M. Cohn, Algebraic Numbers and Algebraic Functions, Chapman and Hall Mathematics Series, Chapman & Hall, London, 1991. doi: 10.1007/978-1-4899-3444-4.
[8]	P. Corvaja and U. Zannier, A lower bound for the height of a rational function at $S$-unit points, Monatsh. Math., 144 (2005), 203-224. doi: 10.1007/s00605-004-0273-0.
[9]	E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith., 34 (1979), 391-401.
[10]	M. Einsiedler and D. Lind, Algebraic $\mathbbZ^d$-actions of entropy rank one, Trans. Amer. Math. Soc., 356 (2004), 1799-1831. doi: 10.1090/S0002-9947-04-03554-8.
[11]	M. Einsiedler, D. Lind, R. Miles and T. Ward, Expansive subdynamics for algebraic $\mathbbZ^d$-actions, Ergodic Theory Dynam. Systems, 21 (2001), 1695-1729. doi: 10.1017/S014338570100181X.
[12]	G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Universitext, Springer-Verlag London, Ltd., London, 1999. doi: 10.1007/978-1-4471-3898-3.
[13]	D. Fried, Entropy for smooth abelian actions, Proc. Amer. Math. Soc., 87 (1983), 111-116. doi: 10.1090/S0002-9939-1983-0677244-7.
[14]	S. Friedland, Entropy of graphs, semigroups and groups}, in Ergodic Theory of $Z^d$ Actions (Warwick, 1993-1994), London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996, 319-343. doi: 10.1017/CBO9780511662812.013.
[15]	W. Geller and M. Pollicott, An entropy for $\mathbb Z^2$-actions with finite entropy generators, Dedicated to the memory of Wiesław Szlenk, Fund. Math., 157 (1998), 209-220.
[16]	A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms, arXiv:1211.0987, to appear.
[17]	A. Katok, S. Katok and F. R. Hertz, The Fried average entropy and slow entropy for actions of higher rank abelian groups, Geometric and Functional Analysis, 24 (2014), 1204-1228. doi: 10.1007/s00039-014-0284-5.
[18]	B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems, 9 (1989), 691-735. doi: 10.1017/S0143385700005290.
[19]	F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A561-A563.
[20]	D. Lind, K. Schmidt and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math., 101 (1990), 593-629. doi: 10.1007/BF01231517.
[21]	D. A. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy, Ergodic Theory Dynam. Systems, 8 (1988), 411-419. doi: 10.1017/S0143385700004545.
[22]	R. Miles, A natural boundary for the dynamical zeta function for commuting group automorphisms, Proc. Amer. Math. Soc., 143 (2015), 2927-2933. doi: 10.1090/S0002-9939-2015-12515-4.
[23]	R. Miles, Zeta functions for elements of entropy rank-one actions, Ergodic Theory Dynam. Systems, 27 (2007), 567-582. doi: 10.1017/S0143385706000794.
[24]	R. Miles, Finitely represented closed-orbit subdynamics for commuting automorphisms, Ergodic Theory Dynam. Systems, 30 (2010), 1787-1802. doi: 10.1017/S0143385709000741.
[25]	R. Miles, Synchronization points and associated dynamical invariants, Trans. Amer. Math. Soc., 365 (2013), 5503-5524. doi: 10.1090/S0002-9947-2013-05829-1.
[26]	R. Miles, M. Staines and T. Ward, Dynamical invariants for group automorphisms, Contemp. Math., 631 (2015), 231-258. doi: 10.1090/conm/631/12606.
[27]	R. Miles and T. Ward, Periodic point data detects subdynamics in entropy rank one, Ergodic Theory Dynam. Systems, 26 (2006), 1913-1930. doi: 10.1017/S014338570600054X.
[28]	R. Miles and T. Ward, Uniform periodic point growth in entropy rank one, Proc. Amer. Math. Soc., 136 (2008), 359-365. doi: 10.1090/S0002-9939-07-09018-1.
[29]	R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc., 137 (2009), 1499-1507. doi: 10.1090/S0002-9939-08-09649-4.
[30]	R. Miles and T. Ward, A dichotomy in orbit growth for commuting automorphisms, J. Lond. Math. Soc. (2), 81 (2010), 715-726. doi: 10.1112/jlms/jdq010.
[31]	R. Miles and T. Ward, A directional uniformity of periodic point distribution and mixing, Discrete Contin. Dyn. Syst., 30 (2011), 1181-1189. doi: 10.3934/dcds.2011.30.1181.
[32]	J. Milnor, On the entropy geometry of cellular automata, Complex Systems, 2 (1988), 357-385.
[33]	G. Morris and T. Ward, Entropy bounds for endomorphisms commuting with $K$ actions, Israel J. Math., 106 (1998), 1-11. doi: 10.1007/BF02773458.
[34]	M. Pollicott, A note on the growth of periodic points for commuting toral automorphisms, ISRN Geometry, 2012 (2012), Article ID 165808, 15 pages. doi: 10.5402/2012/165808.
[35]	K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-0277-2.
[36]	K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei, Invent. Math., 111 (1993), 69-76. doi: 10.1007/BF01231280.
[37]	K. R. Yu, Linear forms in $p$-adic logarithms. II, Compositio Math., 74 (1990), 15-113; Available from: http://www.numdam.org/item?id=CM_1990__74_1_15_0.

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