May  2021, 41(5): 2051-2070. doi: 10.3934/dcds.2020352

Cylinder absolute games on solenoids

1. 

Beijing International Center for Mathematical Research, Peking University, Beijing, 100 871, China

2. 

Current address: Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100 084, China

Received  August 2019 Published  October 2020

Fund Project: Parts of this work first appeared in a slightly different avatar in the author's PhD thesis submitted to the Tata Institute of Fundamental Research, Bombay in 2017. For a portion of that duration, financial support from CSIR, Government of India under SPM-07/858(0199)/2014- EMR-I is duly acknowledged

Let $ A $ be any affine surjective endomorphism of a solenoid ${\Sigma_{{\mathcal{P}}}} $ over the circle $ S^1 $ which is not an infinite-order translation of $ {\Sigma_{{\mathcal{P}}}}$. We prove the existence of a cylinder absolute winning (CAW) subset $ F \subseteq {\Sigma_{{\mathcal{P}}}} $ with the property that for any $ x \in F $, the orbit closure $ \overline{\{ A^{\ell} x \mid \ell \in {\mathbb{N}} \}} $ does not contain any periodic orbits. A measure $ \mu $ on a metric space is said to be Federer if for all small enough balls around any generic point with respect to $ \mu $, the measure grows by at most some constant multiple on doubling the radius of the ball. The class of infinite solenoids considered in this paper provides, to the best of our knowledge, some of the early natural examples of non-Federer spaces where absolute games can be played and won. Dimension maximality and incompressibility of CAW sets is also discussed for a number of possibilities in addition to their winning nature for the games known from before.

Citation: L. Singhal. Cylinder absolute games on solenoids. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2051-2070. doi: 10.3934/dcds.2020352
References:
[1]

J. An, A. Ghosh, L. Guan and T. Ly, Bounded orbits of diagonalizable flows on finite volume quotients of products of $ {\rm {SL}}_2(\mathbb R)$, Adv. Math., 354 (2019), 106743, 18 pp. doi: 10.1016/j.aim.2019.106743.  Google Scholar

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D. BadziahinA. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Ann. of Math. (2), 174 (2011), 1837-1883.  doi: 10.4007/annals.2011.174.3.9.  Google Scholar

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D. Berend, Ergodic semigroups of epimorphisms, Trans. Amer. Math. Soc., 289 (1985), 393-407.  doi: 10.1090/S0002-9947-1985-0779072-7.  Google Scholar

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R. BroderickL. Fishman and D. Kleinbock, Schmidt's game, fractals, and orbits of toral endomorphisms, Ergodic Theory Dynam. Systems, 31 (2011), 1095-1107.  doi: 10.1017/S0143385710000374.  Google Scholar

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R. BroderickL. FishmanD. KleinbockA. Reich and B. Weiss, The set of badly approximable vectors is strongly $C^1$ incompressible., Math. Proc. Cambridge Philos. Soc., 153 (2012), 319-339.  doi: 10.1017/S0305004112000242.  Google Scholar

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S. G. Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv., 61 (1986), 636-660.   Google Scholar

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S. G. Dani, On orbits of endomorphisms of tori and the Schmidt game, Ergodic Theory Dynam. Systems, 8 (1988), 523-529.  doi: 10.1017/S0143385700004673.  Google Scholar

[9]

S. G. Dani, On badly approximable numbers, Schmidt games and bounded orbits of flows, in Number Theory and Dynamical Systems (eds. M. M. Dodson and J. A. G. Vickers), Cambridge Univ. Press, 134 (1989), 69–86. doi: 10.1017/CBO9780511661983.006.  Google Scholar

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K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

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L. Fishman, D. Simmons and M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces, Mem. Amer. Math. Soc., 254 (2018), v+137pp. doi: 10.1090/memo/1215.  Google Scholar

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S. A. Juzvinskiĭ, Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. Ž., 8 (1967), 230–239.  Google Scholar

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D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ 's Moscow Seminar on Dynamical Systems, Amer. Math. Soc., 28 (1996), 141–172. doi: 10.1090/trans2/171/11.  Google Scholar

[14]

D. Kleinbock and T. Ly, Badly approximable $S$-numbers and absolute Schmidt games, J. Number Theory, 164 (2016), 13-42.  doi: 10.1016/j.jnt.2015.12.014.  Google Scholar

[15]

D. Kleinbock and B. Weiss, Modified Schmidt games and Diophantine approximation with weights, Adv. Math., 223 (2010), 1276-1298.  doi: 10.1016/j.aim.2009.09.018.  Google Scholar

[16]

S. Kristensen, Badly approximable systems of linear forms over a field of formal series., J. Théor. Nombres Bordeaux, 18 (2006), 421-444.  doi: 10.5802/jtnb.552.  Google Scholar

[17]

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.  Google Scholar

[18]

C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3.  Google Scholar

[19] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge University Press, Cambridge, 2007.   Google Scholar
[20]

W. M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc., 123 (1966), 178-199.  doi: 10.1090/S0002-9947-1966-0195595-4.  Google Scholar

[21]

S. Semmes, Some remarks about solenoids, 2, preprint, arXiv: 1210.0145. Google Scholar

[22]

S. Weil, Schmidt games and conditions on resonant sets, preprint, arXiv: 1210.1152. Google Scholar

[23]

A. M. Wilson, On endomorphisms of a solenoid, Proc. Amer. Math. Soc., 55 (1976), 69-74.  doi: 10.1090/S0002-9939-1976-0390181-7.  Google Scholar

show all references

References:
[1]

J. An, A. Ghosh, L. Guan and T. Ly, Bounded orbits of diagonalizable flows on finite volume quotients of products of $ {\rm {SL}}_2(\mathbb R)$, Adv. Math., 354 (2019), 106743, 18 pp. doi: 10.1016/j.aim.2019.106743.  Google Scholar

[2]

C. S. Aravinda, Bounded geodesics and Hausdorff dimension, Math. Proc. Cambridge Philos. Soc., 116 (1994), 505-511.  doi: 10.1017/S0305004100072777.  Google Scholar

[3]

D. BadziahinA. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Ann. of Math. (2), 174 (2011), 1837-1883.  doi: 10.4007/annals.2011.174.3.9.  Google Scholar

[4]

D. Berend, Ergodic semigroups of epimorphisms, Trans. Amer. Math. Soc., 289 (1985), 393-407.  doi: 10.1090/S0002-9947-1985-0779072-7.  Google Scholar

[5]

R. BroderickL. Fishman and D. Kleinbock, Schmidt's game, fractals, and orbits of toral endomorphisms, Ergodic Theory Dynam. Systems, 31 (2011), 1095-1107.  doi: 10.1017/S0143385710000374.  Google Scholar

[6]

R. BroderickL. FishmanD. KleinbockA. Reich and B. Weiss, The set of badly approximable vectors is strongly $C^1$ incompressible., Math. Proc. Cambridge Philos. Soc., 153 (2012), 319-339.  doi: 10.1017/S0305004112000242.  Google Scholar

[7]

S. G. Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv., 61 (1986), 636-660.   Google Scholar

[8]

S. G. Dani, On orbits of endomorphisms of tori and the Schmidt game, Ergodic Theory Dynam. Systems, 8 (1988), 523-529.  doi: 10.1017/S0143385700004673.  Google Scholar

[9]

S. G. Dani, On badly approximable numbers, Schmidt games and bounded orbits of flows, in Number Theory and Dynamical Systems (eds. M. M. Dodson and J. A. G. Vickers), Cambridge Univ. Press, 134 (1989), 69–86. doi: 10.1017/CBO9780511661983.006.  Google Scholar

[10]

K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[11]

L. Fishman, D. Simmons and M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces, Mem. Amer. Math. Soc., 254 (2018), v+137pp. doi: 10.1090/memo/1215.  Google Scholar

[12]

S. A. Juzvinskiĭ, Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. Ž., 8 (1967), 230–239.  Google Scholar

[13]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ 's Moscow Seminar on Dynamical Systems, Amer. Math. Soc., 28 (1996), 141–172. doi: 10.1090/trans2/171/11.  Google Scholar

[14]

D. Kleinbock and T. Ly, Badly approximable $S$-numbers and absolute Schmidt games, J. Number Theory, 164 (2016), 13-42.  doi: 10.1016/j.jnt.2015.12.014.  Google Scholar

[15]

D. Kleinbock and B. Weiss, Modified Schmidt games and Diophantine approximation with weights, Adv. Math., 223 (2010), 1276-1298.  doi: 10.1016/j.aim.2009.09.018.  Google Scholar

[16]

S. Kristensen, Badly approximable systems of linear forms over a field of formal series., J. Théor. Nombres Bordeaux, 18 (2006), 421-444.  doi: 10.5802/jtnb.552.  Google Scholar

[17]

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.  Google Scholar

[18]

C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3.  Google Scholar

[19] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge University Press, Cambridge, 2007.   Google Scholar
[20]

W. M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc., 123 (1966), 178-199.  doi: 10.1090/S0002-9947-1966-0195595-4.  Google Scholar

[21]

S. Semmes, Some remarks about solenoids, 2, preprint, arXiv: 1210.0145. Google Scholar

[22]

S. Weil, Schmidt games and conditions on resonant sets, preprint, arXiv: 1210.1152. Google Scholar

[23]

A. M. Wilson, On endomorphisms of a solenoid, Proc. Amer. Math. Soc., 55 (1976), 69-74.  doi: 10.1090/S0002-9939-1976-0390181-7.  Google Scholar

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