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October  2019, 39(10): 5743-5774. doi: 10.3934/dcds.2019252

Product of expansive Markov maps with hole

Haritha C and  Nikita Agarwal ,

Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India

* Corresponding author

Received  August 2018 Revised  March 2019 Published  July 2019

Fund Project: The research of the first author is supported by the Council of Scientific & Industrial Research (CSIR), India (File no. 09/1020(0133)/2018-EMR-I), and the second author is supported by Center for Research on Environment and Sustainable Technologies (CREST), IISER Bhopal, CoE funded by the Ministry of Human Resource Development (MHRD), 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.

Keywords: Open dynamical systems, escape rate, expansive Markov maps, symbolic dynamics, combinatorics.
Mathematics Subject Classification: Primary: 37A05, 37B10.
Citation: Haritha C, Nikita Agarwal. Product of expansive Markov maps with hole. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5743-5774. doi: 10.3934/dcds.2019252
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[16]

M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, London Mathematical Society Student Texts, 1998. doi: 10.1017/CBO9781139173049.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[16]

M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, London Mathematical Society Student Texts, 1998. doi: 10.1017/CBO9781139173049.  Google Scholar

Figure 1.  Examples of expansive Markov maps
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Figure 2.  Basic rectangles when $M = 3$, $N = 2$ and (left) $m = n = 1$; $R_{i,j,1,1}\sim C_{2i+j}$ and (right) $m = n = 2$; (1) $R_{5,2,2,2}\sim C_{34}$, (2) $R_{1,1,2,2}\sim C_{03}$
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Figure 3.  $M = 3,N = 2,m = 2,n = 1$, each rectangle is a union of two basic rectangles: (1) $R_{0,0,2,1} = R_{0,0,2,2}\cup R_{0,1,2,2}\sim C_{00}\cup C_{01}$, (2) $R_{6,1,2,1} = R_{6,2,2,2}\cup R_{6,3,2,2}\sim C_{50}\cup C_{51}$
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Figure 4.  Escape rates into basic rectangles for $M = 3, N = 2$: (left) $m = n = 2$; $\rho(A)\sim 0.025$, $\rho(B)\sim 0.029$; (right) $m = n = 3$; $\rho(A)\sim 0.0039$, $\rho(B)\sim 0.0046$, $\rho(C)\sim 0.0047$. In each square, rectangles with the same color have the same escape rate
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Figure 6.  Escape rates for rectangles when (left) $m = 1,n = 2$; $\rho(A)\sim 0.08$, $\rho(B)\sim 0.1$; (right) $m = 1,n = 3$; $\rho(A)\sim 0.036$, $\rho(B)\sim 0.042$, $\rho(C)\sim 0.047$. In each square, rectangles with the same color have the same escape rate
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Figure 5.  Escape rates of rectangles when $M = 3$, $N = 2$, $m = 2$, $n = 1$, $\rho(A)\sim 0.051$, $\rho(B)\sim 0.061$. Rectangles with the same color have the same escape rate
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Figure 7.  Rectangles corresponding to the collection $S$ in Construction 1 for $m = 3$ and $q = 6$ for the map $T_{3,2}$ on $\mathbb{T}^2$
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Figure 8.  Maps conjugate to shift map on subshift of finite type
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Table 1.  $m = 1,n = 2$, each rectangle corresponds to union of three cylinders based at words of length two. Only two possible values of the escape rate, approximate values are shown
$R_{i,j,m,n}$ Corresponding words $\tau_{\text{min}}$ $\rho(R_{i,j,m,n})\sim$
$R_{0,0,1,2}$ $00,02,04$ 1 0.08
$R_{0,1,1,2}$ $01,03,05$ 2 0.1
$R_{0,2,1,2}$ $10,12,14$ 2 0.1
$R_{0,3,1,2}$ $11,13,15$ 1 0.08
$R_{1,0,1,2}$ $20,22,24$ 1 0.08
$R_{1,1,1,2}$ $21,23,25$ 2 0.1
$R_{1,2,1,2}$ $30,32,34$ 2 0.1
$R_{1,3,1,2}$ $31,33,35$ 1 0.08
$R_{2,0,1,2}$ $40,42,44$ 1 0.08
$R_{2,1,1,2}$ $41,43,45$ 2 0.1
$R_{2,2,1,2}$ $50,52,54$ 2 0.1
$R_{2,3,1,2}$ $51,53,55$ 1 0.08
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$R_{i,j,m,n}$ Corresponding words $\tau_{\text{min}}$ $\rho(R_{i,j,m,n})\sim$
$R_{0,0,1,2}$ $00,02,04$ 1 0.08
$R_{0,1,1,2}$ $01,03,05$ 2 0.1
$R_{0,2,1,2}$ $10,12,14$ 2 0.1
$R_{0,3,1,2}$ $11,13,15$ 1 0.08
$R_{1,0,1,2}$ $20,22,24$ 1 0.08
$R_{1,1,1,2}$ $21,23,25$ 2 0.1
$R_{1,2,1,2}$ $30,32,34$ 2 0.1
$R_{1,3,1,2}$ $31,33,35$ 1 0.08
$R_{2,0,1,2}$ $40,42,44$ 1 0.08
$R_{2,1,1,2}$ $41,43,45$ 2 0.1
$R_{2,2,1,2}$ $50,52,54$ 2 0.1
$R_{2,3,1,2}$ $51,53,55$ 1 0.08
Table 2.  $m = 1,n = 3$, each rectangle corresponds to union of nine cylinders based at words of length three. Only three possible values of the escape rate, approximate values are shown
$R_{i,j,m,n}$ Corresponding words $\tau_{\text{min}}$ $\rho(R_{i,j,m,n})\sim$
$R_{0,0,1,3}$ $000,002,004,020,022,024,040,042,044$ 1 0.036
$R_{0,1,1,3}$ $001,003,005,021,023,025,041,043,045$ 3 0.047
$R_{0,2,1,3}$ $010,012,014,030,032,034,050,052,054$ 2 0.042
$R_{0,3,1,3}$ $011,013,015,031,033,035,051,053,055$ 3 0.047
$R_{0,4,1,3}$ $100,102,104,120,122,124,140,142,144$ 3 0.047
$R_{0,5,1,3}$ $101,103,105,121,123,125,141,143,145$ 2 0.042
$R_{0,6,1,3}$ $110,112,114,130,132,134,150,152,154$ 3 0.047
$R_{0,7,1,3}$ $111,113,115,131,133,135,151,153,155$ 1 0.036
$R_{1,0,1,3}$ $200,202,204,220,222,224,240,242,244$ 1 0.036
$R_{1,1,1,3}$ $201,203,205,221,223,225,241,243,245$ 3 0.047
$R_{1,2,1,3}$ $210,212,214,230,232,234,250,252,254$ 2 0.042
$R_{1,3,1,3}$ $211,213,215,231,233,235,251,253,255$ 3 0.047
$R_{1,4,1,3}$ $300,302,304,320,322,324,340,342,344$ 3 0.047
$R_{1,5,1,3}$ $301,303,305,321,323,325,341,343,345$ 2 0.042
$R_{1,6,1,3}$ $310,312,314,330,332,334,350,352,354$ 3 0.047
$R_{1,7,1,3}$ $311,313,315,331,333,335,351,353,355$ 1 0.036
$R_{2,0,1,3}$ $400,402,404,420,422,424,440,442,444$ 1 0.036
$R_{2,1,1,3}$ $401,403,405,421,423,425,441,443,445$ 3 0.047
$R_{2,2,1,3}$ $410,412,414,430,432,434,450,452,454$ 2 0.042
$R_{2,3,1,3}$ $411,413,415,431,433,435,451,453,455$ 3 0.047
$R_{2,4,1,3}$ $500,502,504,520,522,524,540,542,544$ 3 0.047
$R_{2,5,1,3}$ $501,503,505,521,523,525,541,543,545$ 2 0.042
$R_{2,6,1,3}$ $510,512,514,530,532,534,550,552,554$ 3 0.047
$R_{2,7,1,3}$ $511,513,515,531,533,535,551,553,555$ 1 0.036
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$R_{i,j,m,n}$ Corresponding words $\tau_{\text{min}}$ $\rho(R_{i,j,m,n})\sim$
$R_{0,0,1,3}$ $000,002,004,020,022,024,040,042,044$ 1 0.036
$R_{0,1,1,3}$ $001,003,005,021,023,025,041,043,045$ 3 0.047
$R_{0,2,1,3}$ $010,012,014,030,032,034,050,052,054$ 2 0.042
$R_{0,3,1,3}$ $011,013,015,031,033,035,051,053,055$ 3 0.047
$R_{0,4,1,3}$ $100,102,104,120,122,124,140,142,144$ 3 0.047
$R_{0,5,1,3}$ $101,103,105,121,123,125,141,143,145$ 2 0.042
$R_{0,6,1,3}$ $110,112,114,130,132,134,150,152,154$ 3 0.047
$R_{0,7,1,3}$ $111,113,115,131,133,135,151,153,155$ 1 0.036
$R_{1,0,1,3}$ $200,202,204,220,222,224,240,242,244$ 1 0.036
$R_{1,1,1,3}$ $201,203,205,221,223,225,241,243,245$ 3 0.047
$R_{1,2,1,3}$ $210,212,214,230,232,234,250,252,254$ 2 0.042
$R_{1,3,1,3}$ $211,213,215,231,233,235,251,253,255$ 3 0.047
$R_{1,4,1,3}$ $300,302,304,320,322,324,340,342,344$ 3 0.047
$R_{1,5,1,3}$ $301,303,305,321,323,325,341,343,345$ 2 0.042
$R_{1,6,1,3}$ $310,312,314,330,332,334,350,352,354$ 3 0.047
$R_{1,7,1,3}$ $311,313,315,331,333,335,351,353,355$ 1 0.036
$R_{2,0,1,3}$ $400,402,404,420,422,424,440,442,444$ 1 0.036
$R_{2,1,1,3}$ $401,403,405,421,423,425,441,443,445$ 3 0.047
$R_{2,2,1,3}$ $410,412,414,430,432,434,450,452,454$ 2 0.042
$R_{2,3,1,3}$ $411,413,415,431,433,435,451,453,455$ 3 0.047
$R_{2,4,1,3}$ $500,502,504,520,522,524,540,542,544$ 3 0.047
$R_{2,5,1,3}$ $501,503,505,521,523,525,541,543,545$ 2 0.042
$R_{2,6,1,3}$ $510,512,514,530,532,534,550,552,554$ 3 0.047
$R_{2,7,1,3}$ $511,513,515,531,533,535,551,553,555$ 1 0.036
Table 3.  $m = 4,n = 2$, each rectangle corresponds to union of four cylinders based at words of length four. Only five possible values of the escape rate, approximate values are shown
$R_{i,j,m,n}$ Corresponding words $\tau_{\text{min}}$ $\rho(R_{i,j,m,n})\sim$
$R_{0,0,4,2}$ $0000,0001,0010,0011$ 1 0.0028
$R_{1,0,4,2}$ $0002,0003,0012,0013$ 4 0.00312
$R_{2,0,4,2}$ $0020,0021,0030,0031$ 3 0.00309
$R_{10,0,4,2}$ $0202,0203,0212,0213$ 2 0.00303
$R_{0,1,4,2}$ $0100,0101,0110,0111$ 2 0.00301
$R_{1,1,4,2}$ $0102,0103,0112,0113$ 4 0.00312
$R_{2,1,4,2}$ $0120,0121,0130,0131$ 3 0.00309
$R_{10,1,4,2}$ $0302,0303,0312,0313$ 2 0.00303
$R_{0,2,4,2}$ $1000,1001,1010,1011$ 2 0.00301
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$R_{i,j,m,n}$ Corresponding words $\tau_{\text{min}}$ $\rho(R_{i,j,m,n})\sim$
$R_{0,0,4,2}$ $0000,0001,0010,0011$ 1 0.0028
$R_{1,0,4,2}$ $0002,0003,0012,0013$ 4 0.00312
$R_{2,0,4,2}$ $0020,0021,0030,0031$ 3 0.00309
$R_{10,0,4,2}$ $0202,0203,0212,0213$ 2 0.00303
$R_{0,1,4,2}$ $0100,0101,0110,0111$ 2 0.00301
$R_{1,1,4,2}$ $0102,0103,0112,0113$ 4 0.00312
$R_{2,1,4,2}$ $0120,0121,0130,0131$ 3 0.00309
$R_{10,1,4,2}$ $0302,0303,0312,0313$ 2 0.00303
$R_{0,2,4,2}$ $1000,1001,1010,1011$ 2 0.00301
Table 4.  Upper bound on $m$ in Construction 2 with $q = 6$
$n$ $\ell=1$ $\ell=2$ $\ell=3$ $\ell=4$
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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$n$ $\ell=1$ $\ell=2$ $\ell=3$ $\ell=4$
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
Table 5.  Escape rates for $f$ into holes corresponding to a cylinder based at an allowed word of length two
Holes $H=R_{ij}$ $\tilde{\mu}(H)$ $\rho(H)\sim$ $\tau_{\text{min}}(H)$
$R_{00}$ 0.2 0.188 1
$R_{01}$, $R_{02}$, $R_{10}$, $R_{20}$ 0.124 0.153 2
$R_{03}$, $R_{12}$, $R_{21}$, $R_{30}$ 0.076 0.081 2
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Holes $H=R_{ij}$ $\tilde{\mu}(H)$ $\rho(H)\sim$ $\tau_{\text{min}}(H)$
$R_{00}$ 0.2 0.188 1
$R_{01}$, $R_{02}$, $R_{10}$, $R_{20}$ 0.124 0.153 2
$R_{03}$, $R_{12}$, $R_{21}$, $R_{30}$ 0.076 0.081 2
Table 6.  Escape rates for $f$ into holes corresponding to a cylinder based at an allowed word of length three
Holes $H=R_{ijk}$ $\tilde{\mu}(H)$ $\rho(H)\sim$ $\tau_{\text{min}}(H)$
$R_{010}$, $R_{020}$, $R_{030}$ 0.076 0.081 2
$R_{000}$ 0.076 0.057 1
$R_{001}$, $R_{002}$, $R_{012}$, $R_{021}$, $R_{100}$, $R_{120}$, $R_{200}$, $R_{210}$ 0.047 0.054 3
$R_{003}$, $R_{102}$, $R_{201}$, $R_{300}$ 0.029 0.031 3
$R_{101}$, $R_{121}$, $R_{202}$, $R_{212}$ 0.029 0.028 2
$R_{103}$, $R_{203}$, $R_{301}$, $R_{302}$ 0.018 0.019 3
$R_{303}$ 0.011 0.010 2
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Holes $H=R_{ijk}$ $\tilde{\mu}(H)$ $\rho(H)\sim$ $\tau_{\text{min}}(H)$
$R_{010}$, $R_{020}$, $R_{030}$ 0.076 0.081 2
$R_{000}$ 0.076 0.057 1
$R_{001}$, $R_{002}$, $R_{012}$, $R_{021}$, $R_{100}$, $R_{120}$, $R_{200}$, $R_{210}$ 0.047 0.054 3
$R_{003}$, $R_{102}$, $R_{201}$, $R_{300}$ 0.029 0.031 3
$R_{101}$, $R_{121}$, $R_{202}$, $R_{212}$ 0.029 0.028 2
$R_{103}$, $R_{203}$, $R_{301}$, $R_{302}$ 0.018 0.019 3
$R_{303}$ 0.011 0.010 2
Table 7.  Escape rates for $f = T_2\times T_2\times S$ into holes corresponding to a cylinder based at an allowed word of length two
Holes $H$ $\tilde{\mu}(H)\sim $ $\rho(H)\sim$ $\tau_{\text{min}}(H)$
$R_{aa}$, $a$ is even 0.0279 0.0251 1
$R_{ab}$, exactly one of $a$ or $b$ is even 0.0173 0.0176 2
$R_{ab}$, both $a\ne b$ are even 0.0279 0.0293 2
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Holes $H$ $\tilde{\mu}(H)\sim $ $\rho(H)\sim$ $\tau_{\text{min}}(H)$
$R_{aa}$, $a$ is even 0.0279 0.0251 1
$R_{ab}$, exactly one of $a$ or $b$ is even 0.0173 0.0176 2
$R_{ab}$, both $a\ne b$ are even 0.0279 0.0293 2
Table 8.  Escape rate for $T_2$ into holes corresponding to a cylinder based at an allowed word of length two. Only two possible values of the escape rate, approximate values are shown
Hole $H=I_{ab}$ $\tilde{\mu}(H)\sim$ $a(z)$ $a(3)$ $\rho(H)\sim$
$I_{01},I_{02},I_{10},I_{20}$ 0.1057 $\dfrac{2}{z+1}$ $\dfrac{1}{2}$ 0.1237
$I_{11},I_{22}$ 0.1443 $\dfrac{2}{z+1}$ $\dfrac{1}{2}$ 0.1237
$I_{12},I_{21}$ 0.1443 $\dfrac{2z+1}{z(z+1)}$ $\dfrac{7}{12}$ 0.1955
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Hole $H=I_{ab}$ $\tilde{\mu}(H)\sim$ $a(z)$ $a(3)$ $\rho(H)\sim$
$I_{01},I_{02},I_{10},I_{20}$ 0.1057 $\dfrac{2}{z+1}$ $\dfrac{1}{2}$ 0.1237
$I_{11},I_{22}$ 0.1443 $\dfrac{2}{z+1}$ $\dfrac{1}{2}$ 0.1237
$I_{12},I_{21}$ 0.1443 $\dfrac{2z+1}{z(z+1)}$ $\dfrac{7}{12}$ 0.1955
Table 9.  Escape rate for $T_3$ into holes corresponding to a cylinder based at an allowed word of length two. Only four possible values of the escape rate, approximate values are shown
Hole $H=I_{ab}$ $\tilde{\mu}(H)\sim$ $a(z)$ $a(3)$ $\rho(H)\sim$
$I_{00},I_{22}$ 0.1056 $\dfrac{2}{z+1}$ $\dfrac{1}{2}$ 0.0810
$I_{01},I_{12}$ 0.1708 $\dfrac{2}{z}$ $\dfrac{2}{3}$ 0.2693
$I_{10},I_{21}$ 0.1056 $\dfrac{2z-1}{z^2}$ $\dfrac{5}{9}$ 0.1188
$I_{11}$ 0.1708 $\dfrac{2z+1}{z(z+1)}$ $\dfrac{7}{12}$ 0.1528
$I_{20}$ 0.0652 $\dfrac{2}{z+1}$ $\dfrac{1}{2}$ 0.0810
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Hole $H=I_{ab}$ $\tilde{\mu}(H)\sim$ $a(z)$ $a(3)$ $\rho(H)\sim$
$I_{00},I_{22}$ 0.1056 $\dfrac{2}{z+1}$ $\dfrac{1}{2}$ 0.0810
$I_{01},I_{12}$ 0.1708 $\dfrac{2}{z}$ $\dfrac{2}{3}$ 0.2693
$I_{10},I_{21}$ 0.1056 $\dfrac{2z-1}{z^2}$ $\dfrac{5}{9}$ 0.1188
$I_{11}$ 0.1708 $\dfrac{2z+1}{z(z+1)}$ $\dfrac{7}{12}$ 0.1528
$I_{20}$ 0.0652 $\dfrac{2}{z+1}$ $\dfrac{1}{2}$ 0.0810
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