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On the maximal saddle order of $ p:-q $ resonant saddle
Product of expansive Markov maps with hole
Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India |
We consider product of expansive Markov maps on an interval with hole which is conjugate to a subshift of finite type. For certain class of maps, it is known that the escape rate into a given hole does not just depend on its size but also on its position in the state space. We illustrate this phenomenon for maps considered here. We compare the escape rate into a connected hole and a hole which is a union of holes with a certain property, but have same measure. This gives rise to some interesting combinatorial problems.
References:
[1] |
R. L. Adler,
Symbolic dynamics and Markov Partitions, Bulletin of American Mathematical Society, 35 (1998), 1-56.
doi: 10.1090/S0273-0979-98-00737-X. |
[2] |
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. |
[3] |
W. Bahsoun, C. Bose and G. Froyland (Editors), Ergodic Theory, Open Dynamics, and Coherent Structures, Springer-Verlag, New York, 2014.
doi: 10.1007/978-1-4939-0419-8. |
[4] |
H. van den Bedem and N. Chernov,
Expanding maps of an interval with holes, Ergodic Theory and Dynamical Systems, 22 (2002), 637-654.
doi: 10.1017/S0143385702000329. |
[5] |
H. Bruin, M. Demers and I. Melbourne,
Existence and convergence properties of physical measures for certain dynamical systems with holes, Ergodic Theory and Dynamical Systems, 30 (2010), 687-728.
doi: 10.1017/S0143385709000200. |
[6] |
S. Bundfuss, T. Krueger and S. Troubetzkoy,
Topological and symbolic dynamics for hyperbolic systems with holes, Ergodic Theory and Dynamical Systems, 31 (2011), 1305-1323.
doi: 10.1017/S0143385710000556. |
[7] |
L. Bunimovich and A. Yurchenko,
Where to place a hole to achieve a maximal escape rate, Israel Journal of Mathematics, 182 (2011), 229-252.
doi: 10.1007/s11856-011-0030-8. |
[8] |
N. Chernov and R. Markarian,
Ergodic properties of Anosov maps with rectangular holes, Boletim Sociedade Brasileira Matematica, 28 (1997), 271-314.
doi: 10.1007/BF01233395. |
[9] |
M. Demers,
Markov extensions and conditionally invariant measures for certain logistic maps with small holes, Ergodic Theory and Dynamical Systems, 25 (2005), 1139-1171.
doi: 10.1017/S0143385704000963. |
[10] |
M. Demers and L. S. Young,
Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397.
doi: 10.1088/0951-7715/19/2/008. |
[11] |
M. Demers, P. Wright and L. S. Young,
Escape rates and physically relevant measures for billiards with small holes, Communications in Mathematical Physics, 294 (2010), 253-288.
doi: 10.1007/s00220-009-0941-y. |
[12] |
L. J. Guibas and A. M. Odlyzko,
String overlaps, pattern matching, and non-transitive games, Journal of Combinatorial Theory, Series A, 30 (1981), 183-208.
doi: 10.1016/0097-3165(81)90005-4. |
[13] |
G. Keller and C. Liverani,
Rare events, escape rates and quasistationarity: Some exact formulae, Journal of Statistical Physics, 135 (2009), 519-534.
doi: 10.1007/s10955-009-9747-8. |
[14] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.![]() ![]() ![]() |
[15] |
G. Pianigiani and J. A. Yorke,
Expanding maps on sets which are almost invariant: Decay and chaos, Transactions of AMS, 252 (1979), 351-366.
doi: 10.2307/1998093. |
[16] |
M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, London Mathematical Society Student Texts, 1998.
doi: 10.1017/CBO9781139173049. |
show all references
References:
[1] |
R. L. Adler,
Symbolic dynamics and Markov Partitions, Bulletin of American Mathematical Society, 35 (1998), 1-56.
doi: 10.1090/S0273-0979-98-00737-X. |
[2] |
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. |
[3] |
W. Bahsoun, C. Bose and G. Froyland (Editors), Ergodic Theory, Open Dynamics, and Coherent Structures, Springer-Verlag, New York, 2014.
doi: 10.1007/978-1-4939-0419-8. |
[4] |
H. van den Bedem and N. Chernov,
Expanding maps of an interval with holes, Ergodic Theory and Dynamical Systems, 22 (2002), 637-654.
doi: 10.1017/S0143385702000329. |
[5] |
H. Bruin, M. Demers and I. Melbourne,
Existence and convergence properties of physical measures for certain dynamical systems with holes, Ergodic Theory and Dynamical Systems, 30 (2010), 687-728.
doi: 10.1017/S0143385709000200. |
[6] |
S. Bundfuss, T. Krueger and S. Troubetzkoy,
Topological and symbolic dynamics for hyperbolic systems with holes, Ergodic Theory and Dynamical Systems, 31 (2011), 1305-1323.
doi: 10.1017/S0143385710000556. |
[7] |
L. Bunimovich and A. Yurchenko,
Where to place a hole to achieve a maximal escape rate, Israel Journal of Mathematics, 182 (2011), 229-252.
doi: 10.1007/s11856-011-0030-8. |
[8] |
N. Chernov and R. Markarian,
Ergodic properties of Anosov maps with rectangular holes, Boletim Sociedade Brasileira Matematica, 28 (1997), 271-314.
doi: 10.1007/BF01233395. |
[9] |
M. Demers,
Markov extensions and conditionally invariant measures for certain logistic maps with small holes, Ergodic Theory and Dynamical Systems, 25 (2005), 1139-1171.
doi: 10.1017/S0143385704000963. |
[10] |
M. Demers and L. S. Young,
Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397.
doi: 10.1088/0951-7715/19/2/008. |
[11] |
M. Demers, P. Wright and L. S. Young,
Escape rates and physically relevant measures for billiards with small holes, Communications in Mathematical Physics, 294 (2010), 253-288.
doi: 10.1007/s00220-009-0941-y. |
[12] |
L. J. Guibas and A. M. Odlyzko,
String overlaps, pattern matching, and non-transitive games, Journal of Combinatorial Theory, Series A, 30 (1981), 183-208.
doi: 10.1016/0097-3165(81)90005-4. |
[13] |
G. Keller and C. Liverani,
Rare events, escape rates and quasistationarity: Some exact formulae, Journal of Statistical Physics, 135 (2009), 519-534.
doi: 10.1007/s10955-009-9747-8. |
[14] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.![]() ![]() ![]() |
[15] |
G. Pianigiani and J. A. Yorke,
Expanding maps on sets which are almost invariant: Decay and chaos, Transactions of AMS, 252 (1979), 351-366.
doi: 10.2307/1998093. |
[16] |
M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, London Mathematical Society Student Texts, 1998.
doi: 10.1017/CBO9781139173049. |








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Corresponding words | |||
1 | 0.036 | ||
3 | 0.047 | ||
2 | 0.042 | ||
3 | 0.047 | ||
3 | 0.047 | ||
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1 | 0.036 | ||
1 | 0.036 | ||
3 | 0.047 | ||
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3 | 0.047 | ||
3 | 0.047 | ||
2 | 0.042 | ||
3 | 0.047 | ||
1 | 0.036 | ||
1 | 0.036 | ||
3 | 0.047 | ||
2 | 0.042 | ||
3 | 0.047 | ||
3 | 0.047 | ||
2 | 0.042 | ||
3 | 0.047 | ||
1 | 0.036 |
Corresponding words | |||
1 | 0.036 | ||
3 | 0.047 | ||
2 | 0.042 | ||
3 | 0.047 | ||
3 | 0.047 | ||
2 | 0.042 | ||
3 | 0.047 | ||
1 | 0.036 | ||
1 | 0.036 | ||
3 | 0.047 | ||
2 | 0.042 | ||
3 | 0.047 | ||
3 | 0.047 | ||
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3 | 0.047 | ||
1 | 0.036 | ||
1 | 0.036 | ||
3 | 0.047 | ||
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3 | 0.047 | ||
3 | 0.047 | ||
2 | 0.042 | ||
3 | 0.047 | ||
1 | 0.036 |
Corresponding words | |||
1 | 0.0028 | ||
4 | 0.00312 | ||
3 | 0.00309 | ||
2 | 0.00303 | ||
2 | 0.00301 | ||
4 | 0.00312 | ||
3 | 0.00309 | ||
2 | 0.00303 | ||
2 | 0.00301 |
Corresponding words | |||
1 | 0.0028 | ||
4 | 0.00312 | ||
3 | 0.00309 | ||
2 | 0.00303 | ||
2 | 0.00301 | ||
4 | 0.00312 | ||
3 | 0.00309 | ||
2 | 0.00303 | ||
2 | 0.00301 |
1 | 1 | 2 | 2 | 2 |
2 | 10 | 7 | 5 | 3 |
3 | 20 | 11 | 7 | 5 |
4 | 30 | 15 | 10 | 7 |
5 | 40 | 20 | 12 | 8 |
6 | 50 | 24 | 15 | 10 |
7 | 59 | 29 | 18 | 12 |
8 | 69 | 33 | 20 | 13 |
9 | 79 | 38 | 23 | 15 |
1 | 1 | 2 | 2 | 2 |
2 | 10 | 7 | 5 | 3 |
3 | 20 | 11 | 7 | 5 |
4 | 30 | 15 | 10 | 7 |
5 | 40 | 20 | 12 | 8 |
6 | 50 | 24 | 15 | 10 |
7 | 59 | 29 | 18 | 12 |
8 | 69 | 33 | 20 | 13 |
9 | 79 | 38 | 23 | 15 |
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0.2 | 0.188 | 1 | |
0.124 | 0.153 | 2 | |
0.076 | 0.081 | 2 |
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0.2 | 0.188 | 1 | |
0.124 | 0.153 | 2 | |
0.076 | 0.081 | 2 |
Holes |
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0.076 | 0.081 | 2 | |
0.076 | 0.057 | 1 | |
0.047 | 0.054 | 3 | |
0.029 | 0.031 | 3 | |
0.029 | 0.028 | 2 | |
0.018 | 0.019 | 3 | |
0.011 | 0.010 | 2 |
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0.076 | 0.081 | 2 | |
0.076 | 0.057 | 1 | |
0.047 | 0.054 | 3 | |
0.029 | 0.031 | 3 | |
0.029 | 0.028 | 2 | |
0.018 | 0.019 | 3 | |
0.011 | 0.010 | 2 |
Holes |
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0.0279 | 0.0251 | 1 | |
0.0173 | 0.0176 | 2 | |
0.0279 | 0.0293 | 2 |
Holes |
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0.0279 | 0.0251 | 1 | |
0.0173 | 0.0176 | 2 | |
0.0279 | 0.0293 | 2 |
Hole |
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0.1057 | 0.1237 | |||
0.1443 | 0.1237 | |||
0.1443 | 0.1955 |
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0.1057 | 0.1237 | |||
0.1443 | 0.1237 | |||
0.1443 | 0.1955 |
Hole |
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0.1056 | 0.0810 | |||
0.1708 | 0.2693 | |||
0.1056 | 0.1188 | |||
0.1708 | 0.1528 | |||
0.0652 | 0.0810 |
Hole |
||||
0.1056 | 0.0810 | |||
0.1708 | 0.2693 | |||
0.1056 | 0.1188 | |||
0.1708 | 0.1528 | |||
0.0652 | 0.0810 |
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