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Entropy determination based on the ordinal structure of a dynamical system
Directional uniformities, periodic points, and entropy
1. | Uppsala Universitet, Lägerhyddsvägen 1, Hus 1, 5 och 7, 75106 Uppsala, Sweden |
2. | Durham University, Durham DH1 3LE, United Kingdom |
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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