In this article, we study about the escaping rate of the Sinai-Ruelle-Bowen (SRB) measure through holes of positive measure constructed in the Julia set of hyperbolic rational maps. An explicit formula for the escaping rate through holes of different sizes is obtained and the dependence of this rate on the position of the hole is discussed. Later, the Hausdorff dimension of the survivor set is also computed. For an easier and better understanding, the simple quadratic polynomial restricted on the unit circle is studied.
References
Bunimovich, Y. and Yurchenko, A., Where to place a hole to achieve fastest escape rate, Israel J. Math., 182, (2011), pp 229-252. MR2783972 1236.37004 10.1007/s11856-011-0030-8 Bunimovich, Y. and Yurchenko, A., Where to place a hole to achieve fastest escape rate, Israel J. Math., 182, (2011), pp 229-252. MR2783972 1236.37004 10.1007/s11856-011-0030-8
Cakir, I., Chryssaphinou, O. and Må nsson, M., On a conjecture by Eriksson concerning overlap in strings, Combin. Probab. Comput., 8, (1999), pp 429-440. MR1731978 0946.68111 10.1017/S0963548399003806 Cakir, I., Chryssaphinou, O. and Må nsson, M., On a conjecture by Eriksson concerning overlap in strings, Combin. Probab. Comput., 8, (1999), pp 429-440. MR1731978 0946.68111 10.1017/S0963548399003806
Coelho, Z. and Parry, W., Central limit asymptotics for shifts of finite type, Israel J. Math., 69, (1990), pp 235-249. MR1045376 0701.60026 10.1007/BF02937307 Coelho, Z. and Parry, W., Central limit asymptotics for shifts of finite type, Israel J. Math., 69, (1990), pp 235-249. MR1045376 0701.60026 10.1007/BF02937307
Collet, P., Martínez, S. and Schmitt, B., The Pianigiani - Yorke measure for topological Markov chains, Israel J. Math., 97, (1997), pp 61-70. MR1441238 0902.58016 10.1007/BF02774026 Collet, P., Martínez, S. and Schmitt, B., The Pianigiani - Yorke measure for topological Markov chains, Israel J. Math., 97, (1997), pp 61-70. MR1441238 0902.58016 10.1007/BF02774026
Denker, M., Przytycki, F. and Urbanski, M., On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems, 16, (1996), pp 255-266. MR1389624 0852.46024 10.1017/S0143385700008804 Denker, M., Przytycki, F. and Urbanski, M., On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems, 16, (1996), pp 255-266. MR1389624 0852.46024 10.1017/S0143385700008804
Denker, M. and Urbanski, M., Ergodic theory of equilibrium states for rational maps, Nonlinearity, 4, (1991), pp 103-134. MR1092887 0718.58035 10.1088/0951-7715/4/1/008 Denker, M. and Urbanski, M., Ergodic theory of equilibrium states for rational maps, Nonlinearity, 4, (1991), pp 103-134. MR1092887 0718.58035 10.1088/0951-7715/4/1/008
Eriksson, K., Autocorrelation and the enumeration of strings avoiding a fixed string, Combin. Probab. Comput., 6, (1997), pp 45-48. MR1436718 0865.68097 10.1017/S0963548397002836 Eriksson, K., Autocorrelation and the enumeration of strings avoiding a fixed string, Combin. Probab. Comput., 6, (1997), pp 45-48. MR1436718 0865.68097 10.1017/S0963548397002836
Guibas, L. J. and Odlyzko, A. M., String overlaps, pattern matching and non-transitive games, J. Combin. Theory Ser. A, 30, (1981), pp 183-208. MR611250 0454.68109 10.1016/0097-3165(81)90005-4 Guibas, L. J. and Odlyzko, A. M., String overlaps, pattern matching and non-transitive games, J. Combin. Theory Ser. A, 30, (1981), pp 183-208. MR611250 0454.68109 10.1016/0097-3165(81)90005-4
Haydn, N., Convergence of the transfer operator for rational maps, Ergodic Theory Dynam. Systems, 19, (1999), pp 657-669. MR1695914 0953.37006 10.1017/S0143385799130190 Haydn, N., Convergence of the transfer operator for rational maps, Ergodic Theory Dynam. Systems, 19, (1999), pp 657-669. MR1695914 0953.37006 10.1017/S0143385799130190
Hirata, M., Poisson law for Axiom A diffeomorphisms, Ergodic Theory Dynam. Systems, 13, (1993), pp 533-556. MR1245828 10.1017/S0143385700007513 Hirata, M., Poisson law for Axiom A diffeomorphisms, Ergodic Theory Dynam. Systems, 13, (1993), pp 533-556. MR1245828 10.1017/S0143385700007513
Keller, G., Generalised bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69, (1985), pp 461-478. MR787608 Keller, G., Generalised bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69, (1985), pp 461-478. MR787608
Keller, G., and Liverani, C., Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28, (1999), pp 141-152. MR1679080 0956.37003 Keller, G., and Liverani, C., Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28, (1999), pp 141-152. MR1679080 0956.37003
Keller, G. and Liverani, C., Rare events, escape rates and quasistationarity: some exact formulae, J. Stat. Phys., 135, (2009), pp 519-534. MR2535206 1179.37010 10.1007/s10955-009-9747-8 Keller, G. and Liverani, C., Rare events, escape rates and quasistationarity: some exact formulae, J. Stat. Phys., 135, (2009), pp 519-534. MR2535206 1179.37010 10.1007/s10955-009-9747-8
Manning, A., The dimension of maximal measure for a polynomial map, Ann. of Math., 119, (1984), pp 425-430. MR740898 0551.30021 10.2307/2007044 Manning, A., The dimension of maximal measure for a polynomial map, Ann. of Math., 119, (1984), pp 425-430. MR740898 0551.30021 10.2307/2007044
Parry, W. and Pollicott, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque, (1990). MR1085356 0726.58003 Parry, W. and Pollicott, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque, (1990). MR1085356 0726.58003
Pianigiani, G. and Yorke, J., Expanding maps on sets which are almost invariant: decay and chaos, Trans. Amer. Math. Soc., 252, (1979), pp 351-366. MR534126 0417.28010 Pianigiani, G. and Yorke, J., Expanding maps on sets which are almost invariant: decay and chaos, Trans. Amer. Math. Soc., 252, (1979), pp 351-366. MR534126 0417.28010
Pollicott, M. and Sharp, R., Rates of recurrence for $\mathbb{Z}^{q}$ and $\mathbb{R}^{q}$ extensions of subshifts of finite type, J. Lond. Math. Soc., 2, (1994), pp 401-416. MR1260121 0797.60022 10.1112/jlms/49.2.401 Pollicott, M. and Sharp, R., Rates of recurrence for $\mathbb{Z}^{q}$ and $\mathbb{R}^{q}$ extensions of subshifts of finite type, J. Lond. Math. Soc., 2, (1994), pp 401-416. MR1260121 0797.60022 10.1112/jlms/49.2.401
Ruelle, D., Repellers for real analytic maps, Ergodic Theory Dynam. Systems, 2, (1982), pp 99-107. MR684247 0506.58024 10.1017/S0143385700009603 Ruelle, D., Repellers for real analytic maps, Ergodic Theory Dynam. Systems, 2, (1982), pp 99-107. MR684247 0506.58024 10.1017/S0143385700009603
Sridharan, S., Non-vanishing derivatives of Lyapunov exponents and the pressure function, Dyn. Syst., 21, (2006), pp 491-500. MR2273690 10.1080/14689360600872037 Sridharan, S., Non-vanishing derivatives of Lyapunov exponents and the pressure function, Dyn. Syst., 21, (2006), pp 491-500. MR2273690 10.1080/14689360600872037