American Institute of Mathematical Sciences

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

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

* 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.

Citation: Haritha C, Nikita Agarwal. Product of expansive Markov maps with hole. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5743-5774. doi: 10.3934/dcds.2019252
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Examples of expansive Markov maps
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) $R_{1,1,2,2}\sim C_{03}$
$M = 3,N = 2,m = 2,n = 1$, 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) $R_{6,1,2,1} = R_{6,2,2,2}\cup R_{6,3,2,2}\sim C_{50}\cup C_{51}$
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
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
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
Rectangles corresponding to the collection $S$ in Construction 1 for $m = 3$ and $q = 6$ for the map $T_{3,2}$ on $\mathbb{T}^2$
Maps conjugate to shift map on subshift of finite type
$m = 1,n = 2$, 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
 $R_{i,j,m,n}$ 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
$m = 1,n = 3$, 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
 $R_{i,j,m,n}$ 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
$m = 4,n = 2$, 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
 $R_{i,j,m,n}$ 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
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
 $n$ $\ell=1$ $\ell=2$ $\ell=3$ $\ell=4$ 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
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
 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
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
 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
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
 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
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
 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
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
 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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