# American Institute of Mathematical Sciences

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

## 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

Received  November 2014 Revised  March 2015 Published  September 2015

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.
Citation: Richard Miles, Thomas Ward. Directional uniformities, periodic points, and entropy. Discrete & 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. Google Scholar [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. doi: 10.4064/aa113-1-3. Google Scholar [3] A. Baker, Transcendental Number Theory,, 2nd edition, (1990). doi: 10.1017/CBO9780511565977. Google Scholar [4] P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle,, Acta Arith., 18 (1971), 355. Google Scholar [5] M. Boyle and D. Lind, Expansive subdynamics,, Trans. Amer. Math. Soc., 349 (1997), 55. doi: 10.1090/S0002-9947-97-01634-6. Google Scholar [6] V. Chothi, G. Everest and T. Ward, $S$-integer dynamical systems: Periodic points,, J. Reine Angew. Math., 489 (1997), 99. doi: 10.1515/crll.1997.489.99. Google Scholar [7] P. M. Cohn, Algebraic Numbers and Algebraic Functions,, Chapman and Hall Mathematics Series, (1991). doi: 10.1007/978-1-4899-3444-4. Google Scholar [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. doi: 10.1007/s00605-004-0273-0. Google Scholar [9] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial,, Acta Arith., 34 (1979), 391. Google Scholar [10] M. Einsiedler and D. Lind, Algebraic $\mathbbZ^d$-actions of entropy rank one,, Trans. Amer. Math. Soc., 356 (2004), 1799. doi: 10.1090/S0002-9947-04-03554-8. Google Scholar [11] M. Einsiedler, D. Lind, R. Miles and T. Ward, Expansive subdynamics for algebraic $\mathbbZ^d$-actions,, Ergodic Theory Dynam. Systems, 21 (2001), 1695. doi: 10.1017/S014338570100181X. Google Scholar [12] G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics,, Universitext, (1999). doi: 10.1007/978-1-4471-3898-3. Google Scholar [13] D. Fried, Entropy for smooth abelian actions,, Proc. Amer. Math. Soc., 87 (1983), 111. doi: 10.1090/S0002-9939-1983-0677244-7. Google Scholar [14] S. Friedland, Entropy of graphs, semigroups and groups},, in Ergodic Theory of $Z^d$ Actions (Warwick, (1996), 1993. doi: 10.1017/CBO9780511662812.013. Google Scholar [15] W. Geller and M. Pollicott, An entropy for $\mathbb Z^2$-actions with finite entropy generators,, Dedicated to the memory of Wiesław Szlenk, 157 (1998), 209. Google Scholar [16] A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms,, , (). Google Scholar [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. doi: 10.1007/s00039-014-0284-5. Google Scholar [18] B. Kitchens and K. Schmidt, Automorphisms of compact groups,, Ergodic Theory Dynam. Systems, 9 (1989), 691. doi: 10.1017/S0143385700005290. Google Scholar [19] F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant,, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978). Google Scholar [20] D. Lind, K. Schmidt and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups,, Invent. Math., 101 (1990), 593. doi: 10.1007/BF01231517. Google Scholar [21] D. A. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy,, Ergodic Theory Dynam. Systems, 8 (1988), 411. doi: 10.1017/S0143385700004545. Google Scholar [22] R. Miles, A natural boundary for the dynamical zeta function for commuting group automorphisms,, Proc. Amer. Math. Soc., 143 (2015), 2927. doi: 10.1090/S0002-9939-2015-12515-4. Google Scholar [23] R. Miles, Zeta functions for elements of entropy rank-one actions,, Ergodic Theory Dynam. Systems, 27 (2007), 567. doi: 10.1017/S0143385706000794. Google Scholar [24] R. Miles, Finitely represented closed-orbit subdynamics for commuting automorphisms,, Ergodic Theory Dynam. Systems, 30 (2010), 1787. doi: 10.1017/S0143385709000741. Google Scholar [25] R. Miles, Synchronization points and associated dynamical invariants,, Trans. Amer. Math. Soc., 365 (2013), 5503. doi: 10.1090/S0002-9947-2013-05829-1. Google Scholar [26] R. Miles, M. Staines and T. Ward, Dynamical invariants for group automorphisms,, Contemp. Math., 631 (2015), 231. doi: 10.1090/conm/631/12606. Google Scholar [27] R. Miles and T. Ward, Periodic point data detects subdynamics in entropy rank one,, Ergodic Theory Dynam. Systems, 26 (2006), 1913. doi: 10.1017/S014338570600054X. Google Scholar [28] R. Miles and T. Ward, Uniform periodic point growth in entropy rank one,, Proc. Amer. Math. Soc., 136 (2008), 359. doi: 10.1090/S0002-9939-07-09018-1. Google Scholar [29] R. Miles and T. Ward, Orbit-counting for nilpotent group shifts,, Proc. Amer. Math. Soc., 137 (2009), 1499. doi: 10.1090/S0002-9939-08-09649-4. Google Scholar [30] R. Miles and T. Ward, A dichotomy in orbit growth for commuting automorphisms,, J. Lond. Math. Soc. (2), 81 (2010), 715. doi: 10.1112/jlms/jdq010. Google Scholar [31] R. Miles and T. Ward, A directional uniformity of periodic point distribution and mixing,, Discrete Contin. Dyn. Syst., 30 (2011), 1181. doi: 10.3934/dcds.2011.30.1181. Google Scholar [32] J. Milnor, On the entropy geometry of cellular automata,, Complex Systems, 2 (1988), 357. Google Scholar [33] G. Morris and T. Ward, Entropy bounds for endomorphisms commuting with $K$ actions,, Israel J. Math., 106 (1998), 1. doi: 10.1007/BF02773458. Google Scholar [34] M. Pollicott, A note on the growth of periodic points for commuting toral automorphisms,, ISRN Geometry, 2012 (2012). doi: 10.5402/2012/165808. Google Scholar [35] K. Schmidt, Dynamical Systems of Algebraic Origin,, Progress in Mathematics, (1995). doi: 10.1007/978-3-0348-0277-2. Google Scholar [36] K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei,, Invent. Math., 111 (1993), 69. doi: 10.1007/BF01231280. Google Scholar [37] K. R. Yu, Linear forms in $p$-adic logarithms. II,, Compositio Math., 74 (1990), 15. Google Scholar

show all references

##### References:
 [1] L. M. Abramov, The entropy of an automorphism of a solenoidal group,, Teor. Veroyatnost. i Primenen, 4 (1959), 249. Google Scholar [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. doi: 10.4064/aa113-1-3. Google Scholar [3] A. Baker, Transcendental Number Theory,, 2nd edition, (1990). doi: 10.1017/CBO9780511565977. Google Scholar [4] P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle,, Acta Arith., 18 (1971), 355. Google Scholar [5] M. Boyle and D. Lind, Expansive subdynamics,, Trans. Amer. Math. Soc., 349 (1997), 55. doi: 10.1090/S0002-9947-97-01634-6. Google Scholar [6] V. Chothi, G. Everest and T. Ward, $S$-integer dynamical systems: Periodic points,, J. Reine Angew. Math., 489 (1997), 99. doi: 10.1515/crll.1997.489.99. Google Scholar [7] P. M. Cohn, Algebraic Numbers and Algebraic Functions,, Chapman and Hall Mathematics Series, (1991). doi: 10.1007/978-1-4899-3444-4. Google Scholar [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. doi: 10.1007/s00605-004-0273-0. Google Scholar [9] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial,, Acta Arith., 34 (1979), 391. Google Scholar [10] M. Einsiedler and D. Lind, Algebraic $\mathbbZ^d$-actions of entropy rank one,, Trans. Amer. Math. Soc., 356 (2004), 1799. doi: 10.1090/S0002-9947-04-03554-8. Google Scholar [11] M. Einsiedler, D. Lind, R. Miles and T. Ward, Expansive subdynamics for algebraic $\mathbbZ^d$-actions,, Ergodic Theory Dynam. Systems, 21 (2001), 1695. doi: 10.1017/S014338570100181X. Google Scholar [12] G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics,, Universitext, (1999). doi: 10.1007/978-1-4471-3898-3. Google Scholar [13] D. Fried, Entropy for smooth abelian actions,, Proc. Amer. Math. Soc., 87 (1983), 111. doi: 10.1090/S0002-9939-1983-0677244-7. Google Scholar [14] S. Friedland, Entropy of graphs, semigroups and groups},, in Ergodic Theory of $Z^d$ Actions (Warwick, (1996), 1993. doi: 10.1017/CBO9780511662812.013. Google Scholar [15] W. Geller and M. Pollicott, An entropy for $\mathbb Z^2$-actions with finite entropy generators,, Dedicated to the memory of Wiesław Szlenk, 157 (1998), 209. Google Scholar [16] A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms,, , (). Google Scholar [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. doi: 10.1007/s00039-014-0284-5. Google Scholar [18] B. Kitchens and K. Schmidt, Automorphisms of compact groups,, Ergodic Theory Dynam. Systems, 9 (1989), 691. doi: 10.1017/S0143385700005290. Google Scholar [19] F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant,, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978). Google Scholar [20] D. Lind, K. Schmidt and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups,, Invent. Math., 101 (1990), 593. doi: 10.1007/BF01231517. Google Scholar [21] D. A. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy,, Ergodic Theory Dynam. Systems, 8 (1988), 411. doi: 10.1017/S0143385700004545. Google Scholar [22] R. Miles, A natural boundary for the dynamical zeta function for commuting group automorphisms,, Proc. Amer. Math. Soc., 143 (2015), 2927. doi: 10.1090/S0002-9939-2015-12515-4. Google Scholar [23] R. Miles, Zeta functions for elements of entropy rank-one actions,, Ergodic Theory Dynam. Systems, 27 (2007), 567. doi: 10.1017/S0143385706000794. Google Scholar [24] R. Miles, Finitely represented closed-orbit subdynamics for commuting automorphisms,, Ergodic Theory Dynam. Systems, 30 (2010), 1787. doi: 10.1017/S0143385709000741. Google Scholar [25] R. Miles, Synchronization points and associated dynamical invariants,, Trans. Amer. Math. Soc., 365 (2013), 5503. doi: 10.1090/S0002-9947-2013-05829-1. Google Scholar [26] R. Miles, M. Staines and T. Ward, Dynamical invariants for group automorphisms,, Contemp. Math., 631 (2015), 231. doi: 10.1090/conm/631/12606. Google Scholar [27] R. Miles and T. Ward, Periodic point data detects subdynamics in entropy rank one,, Ergodic Theory Dynam. Systems, 26 (2006), 1913. doi: 10.1017/S014338570600054X. Google Scholar [28] R. Miles and T. Ward, Uniform periodic point growth in entropy rank one,, Proc. Amer. Math. Soc., 136 (2008), 359. doi: 10.1090/S0002-9939-07-09018-1. Google Scholar [29] R. Miles and T. Ward, Orbit-counting for nilpotent group shifts,, Proc. Amer. Math. Soc., 137 (2009), 1499. doi: 10.1090/S0002-9939-08-09649-4. Google Scholar [30] R. Miles and T. Ward, A dichotomy in orbit growth for commuting automorphisms,, J. Lond. Math. Soc. (2), 81 (2010), 715. doi: 10.1112/jlms/jdq010. Google Scholar [31] R. Miles and T. Ward, A directional uniformity of periodic point distribution and mixing,, Discrete Contin. Dyn. Syst., 30 (2011), 1181. doi: 10.3934/dcds.2011.30.1181. Google Scholar [32] J. Milnor, On the entropy geometry of cellular automata,, Complex Systems, 2 (1988), 357. Google Scholar [33] G. Morris and T. Ward, Entropy bounds for endomorphisms commuting with $K$ actions,, Israel J. Math., 106 (1998), 1. doi: 10.1007/BF02773458. Google Scholar [34] M. Pollicott, A note on the growth of periodic points for commuting toral automorphisms,, ISRN Geometry, 2012 (2012). doi: 10.5402/2012/165808. Google Scholar [35] K. Schmidt, Dynamical Systems of Algebraic Origin,, Progress in Mathematics, (1995). doi: 10.1007/978-3-0348-0277-2. Google Scholar [36] K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei,, Invent. Math., 111 (1993), 69. doi: 10.1007/BF01231280. Google Scholar [37] K. R. Yu, Linear forms in $p$-adic logarithms. II,, Compositio Math., 74 (1990), 15. Google Scholar
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