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November  2018, 38(11): 5883-5895. doi: 10.3934/dcds.2018255

Open maps: Small and large holes with unusual properties

1. 

Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

2. 

School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

* Corresponding author: Kevin G. Hare

Received  February 2018 Revised  June 2018 Published  August 2018

Fund Project: Research of K. G. Hare was supported by NSERC Grant RGPIN-2014-03154.

Let X be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in X. We show that there exists an arbitrarily small finite overlapping union of shifted cylinders which intersects every orbit under the shift map.

We also show that for any proper subshift Y of X there exists a finite overlapping unions of shifted cylinders such that its survivor set contains Y (in particular, it can have entropy arbitrarily close to the entropy of X). Both results may be seen as somewhat counter-intuitive.

Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.

Citation: Kevin G. Hare, Nikita Sidorov. Open maps: Small and large holes with unusual properties. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5883-5895. doi: 10.3934/dcds.2018255
References:
[1]

V. S. Afraimovich and L. A. Bunimovich, Which hole is leaking the most: A topological approach to study open systems, Nonlinearity, 23 (2010), 643-656.  doi: 10.1088/0951-7715/23/3/012.

[2]

R. Alcaraz Barrera, Topological and ergodic properties of symmetric sub-shifts, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 4459-4486.  doi: 10.3934/dcds.2014.34.4459.

[3]

O. F. Bandtlow, O. Jenkinson and M. Pollicott, Periodic points, escape rates and escape measures, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Math. Stat. 70 (2014), 41–58. doi: 10.1007/978-1-4939-0419-8_3.

[4]

A. Bertrand-Mathis, Développement en base θ, répartition modulo un de la suite (n)n≥0; langages codés et θ-shift, Bull. Soc. Math. Fr., 114 (1986), 271-323. 

[5]

L. A. Bunimovich and A. Yurchenko, Where to place a hole to achieve a maximal escape rate, Israel J. Math., 182 (2011), 229-252.  doi: 10.1007/s11856-011-0030-8.

[6]

J.-M. ChamparnaudG. Hansel and D. Perrin, Unavoidable sets of constant length, Internat. J. Algebra Comput., 14 (2004), 241-251.  doi: 10.1142/S0218196704001700.

[7]

L. Clark, β-transformation with a hole, Discr. Cont. Dyn. Sys. A, 36 (2016), 1249-1269.  doi: 10.3934/dcds.2016.36.1249.

[8]

L. ClarkK. G. Hare and N. Sidorov, The baker's map with a convex hole, Nonlinearity, 31 (2018), 3174-3202.  doi: 10.1088/1361-6544/aab595.

[9]

M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397.  doi: 10.1088/0951-7715/19/2/008.

[10]

M. F. Demers, Dispersing billiards with small holes, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Math. Stat. 70 (2014), 137–170. doi: 10.1007/978-1-4939-0419-8_8.

[11]

A. Ferguson and M. Pollicott, Escape rates for Gibbs measures, Ergodic Theory Dynam. Systems, 32 (2012), 961-988.  doi: 10.1017/S0143385711000058.

[12]

P. Glendinning and N. Sidorov, The doubling map with asymmetrical holes, Ergodic Theory Dynam. Systems, 35 (2015), 1208-1228.  doi: 10.1017/etds.2013.98.

[13]

W.-G. Hu and S.-S. Lin, The natural measure of a symbolic dynamical system, arXiv: 1308.2996

[14]

S. Le Borgne, Un codage sofique des automorphismes hyperboliques du tore, Bol. Soc. Bras. Mat., 30 (1999), 61-93.  doi: 10.1007/BF01235675.

[15]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. doi: 10.1017/CBO9780511626302.

[16]

J. Mykkeltveit, A proof of Golomb's conjecture for the de Bruijn graph, J. Combin. Theory B, 13 (1972), 40-45.  doi: 10.1016/0095-8956(72)90006-8.

[17]

W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung., 11 (1960), 401-416.  doi: 10.1007/BF02020954.

[18]

G. Pianigiani and J. A. Yorke, Expanding maps on sets which are almost invariant. Decay and chaos, Trans. Amer. Math. Soc., 252 (1979), 351-366.  doi: 10.2307/1998093.

[19]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., 8 (1957), 477-493.  doi: 10.1007/BF02020331.

[20]

K. Schmidt, Algebraic codings of expansive group automorphisms and two-sided beta-shifts, Monatsh. Math., 129 (2000), 37-61.  doi: 10.1007/s006050050005.

[21]

M.-P. Schützenberger, On the synchronizing properties of certain prefix codes, Inf. Contr., 7 (1964), 23-36.  doi: 10.1016/S0019-9958(64)90232-3.

[22]

N. Sidorov, Arithmetic Dynamics, in Topics in Dynamics and Ergodic Theory, LMS Lecture Notes Ser. 310 (2003), 145–189. doi: 10.1017/CBO9780511546716.010.

[23]

N. Sidorov, Bijective and general arithmetic codings for Pisot toral automorphisms, J. Dynam. Control Systems, 7 (2001), 447-472.  doi: 10.1023/A:1013104016392.

[24]

N. Sidorov and A. Vershik, Ergodic properties of Erdös measure, the entropy of the goldenshift, and related problems, Monatsh. Math., 126 (1998), 215-261.  doi: 10.1007/BF01367764.

[25]

N. Sidorov and A. Vershik, Bijective arithmetic codings of hyperbolic automorphisms of the 2-torus, and binary quadratic forms, J. Dynam. Control Systems, 4 (1998), 365-399.  doi: 10.1023/A:1022836500100.

[26]

B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269.  doi: 10.1090/S0002-9904-1970-12632-5.

show all references

References:
[1]

V. S. Afraimovich and L. A. Bunimovich, Which hole is leaking the most: A topological approach to study open systems, Nonlinearity, 23 (2010), 643-656.  doi: 10.1088/0951-7715/23/3/012.

[2]

R. Alcaraz Barrera, Topological and ergodic properties of symmetric sub-shifts, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 4459-4486.  doi: 10.3934/dcds.2014.34.4459.

[3]

O. F. Bandtlow, O. Jenkinson and M. Pollicott, Periodic points, escape rates and escape measures, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Math. Stat. 70 (2014), 41–58. doi: 10.1007/978-1-4939-0419-8_3.

[4]

A. Bertrand-Mathis, Développement en base θ, répartition modulo un de la suite (n)n≥0; langages codés et θ-shift, Bull. Soc. Math. Fr., 114 (1986), 271-323. 

[5]

L. A. Bunimovich and A. Yurchenko, Where to place a hole to achieve a maximal escape rate, Israel J. Math., 182 (2011), 229-252.  doi: 10.1007/s11856-011-0030-8.

[6]

J.-M. ChamparnaudG. Hansel and D. Perrin, Unavoidable sets of constant length, Internat. J. Algebra Comput., 14 (2004), 241-251.  doi: 10.1142/S0218196704001700.

[7]

L. Clark, β-transformation with a hole, Discr. Cont. Dyn. Sys. A, 36 (2016), 1249-1269.  doi: 10.3934/dcds.2016.36.1249.

[8]

L. ClarkK. G. Hare and N. Sidorov, The baker's map with a convex hole, Nonlinearity, 31 (2018), 3174-3202.  doi: 10.1088/1361-6544/aab595.

[9]

M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397.  doi: 10.1088/0951-7715/19/2/008.

[10]

M. F. Demers, Dispersing billiards with small holes, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Math. Stat. 70 (2014), 137–170. doi: 10.1007/978-1-4939-0419-8_8.

[11]

A. Ferguson and M. Pollicott, Escape rates for Gibbs measures, Ergodic Theory Dynam. Systems, 32 (2012), 961-988.  doi: 10.1017/S0143385711000058.

[12]

P. Glendinning and N. Sidorov, The doubling map with asymmetrical holes, Ergodic Theory Dynam. Systems, 35 (2015), 1208-1228.  doi: 10.1017/etds.2013.98.

[13]

W.-G. Hu and S.-S. Lin, The natural measure of a symbolic dynamical system, arXiv: 1308.2996

[14]

S. Le Borgne, Un codage sofique des automorphismes hyperboliques du tore, Bol. Soc. Bras. Mat., 30 (1999), 61-93.  doi: 10.1007/BF01235675.

[15]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. doi: 10.1017/CBO9780511626302.

[16]

J. Mykkeltveit, A proof of Golomb's conjecture for the de Bruijn graph, J. Combin. Theory B, 13 (1972), 40-45.  doi: 10.1016/0095-8956(72)90006-8.

[17]

W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung., 11 (1960), 401-416.  doi: 10.1007/BF02020954.

[18]

G. Pianigiani and J. A. Yorke, Expanding maps on sets which are almost invariant. Decay and chaos, Trans. Amer. Math. Soc., 252 (1979), 351-366.  doi: 10.2307/1998093.

[19]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., 8 (1957), 477-493.  doi: 10.1007/BF02020331.

[20]

K. Schmidt, Algebraic codings of expansive group automorphisms and two-sided beta-shifts, Monatsh. Math., 129 (2000), 37-61.  doi: 10.1007/s006050050005.

[21]

M.-P. Schützenberger, On the synchronizing properties of certain prefix codes, Inf. Contr., 7 (1964), 23-36.  doi: 10.1016/S0019-9958(64)90232-3.

[23]

N. Sidorov, Bijective and general arithmetic codings for Pisot toral automorphisms, J. Dynam. Control Systems, 7 (2001), 447-472.  doi: 10.1023/A:1013104016392.

[24]

N. Sidorov and A. Vershik, Ergodic properties of Erdös measure, the entropy of the goldenshift, and related problems, Monatsh. Math., 126 (1998), 215-261.  doi: 10.1007/BF01367764.

[25]

N. Sidorov and A. Vershik, Bijective arithmetic codings of hyperbolic automorphisms of the 2-torus, and binary quadratic forms, J. Dynam. Control Systems, 4 (1998), 365-399.  doi: 10.1023/A:1022836500100.

[26]

B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269.  doi: 10.1090/S0002-9904-1970-12632-5.

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