Abstract
It is well known that open dynamical systems can admit an uncountable number of (absolutely continuous) conditionally invariant measures (ACCIMs) for each prescribed escape rate. We propose and illustrate a convex optimisation-based selection scheme (essentially maximum entropy) for gaining numerical access to some of these measures. The work is similar to the maximum entropy (MAXENT) approach for calculating absolutely continuous invariant measures of nonsingular dynamical systems but contains some interesting new twists, including the following: (i) the natural escape rate is not known in advance, which can destroy convex structure in the problem; (ii) exploitation of convex duality to solve each approximation step induces important (but dynamically relevant and not at first apparent) localisation of support; and (iii) significant potential for application to the approximation of other dynamically interesting objects (e.g. invariant manifolds).
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Notes
- 1.
The motivation in [22, p. 353] went beyond interval maps, including preturbulent phenomena in the now famous Lorenz equations, and metastable structures in atmospheric and other fluid flows and complex systems.
- 2.
In the sense of positive m-measure
- 3.
Clearly m ∘ T −1 ≪ m so that \(T: (A,m\vert _{A}) \rightarrow (X,m)\) is a nonsingular transformation, but \(T: (A,m\vert _{A}) \rightarrow (X,m\vert _{A})\) fails to be nonsingular, as \(m\vert _{A} \circ {T}^{-1}(H_{0}) = m(A \cap {T}^{-1}(H_{0})) > 0\) while \(m\vert _{A}(H_{0}) = 0\).
- 4.
The flexibility to tune α without impact on numerical effort is reminiscent of the use of Ulam’s method to calculate the topological pressure of piecewise smooth dynamical systems by varying an inverse temperature parameter [16].
- 5.
Note that \(\hat{C}_{ki} = 0\forall k\) only if \(B_{i} \cap \hat{ A} = \varnothing \). In this case also each \(\hat{C}_{ij} =\hat{ c}_{i} = 0\) and the value of \({\mathbb{M}}^{{\ast}}\lambda\) on B i is irrelevant to the solution of (P n, α ) (by Lemma 5.5). The function Ψ can be defined to be 1 on such coordinates.
- 6.
A similar argument works if i is such that \([\varPsi (\mathbf{x}_{{\ast}})]_{i} = y_{i}^{{\ast}}/V (\mathbf{x}_{{\ast}})\).
References
Afraimovich, V.S., Bunimovich, L.A.: Which hole is leaking the most: a topological approach to study open systems. Nonlinearity 23(3), 643–656 (2010). doi:10.1088/0951-7715/23/3/012
Bahsoun, W., Vaienti, S.: Metastability of certain intermittent maps. Nonlinearity 25(1), 107–124 (2012). doi:10.1088/0951-7715/25/1/107
Borwein, J., Lewis, A.: Duality relationships for entropy-like minimization problems. SIAM J. Control. Optim. 26, 325–338 (1991)
Bose, C.J., Murray, R.: Minimum ‘energy’ approximations of invariant measures for nonsingular transformations. Discrete Contin. Dyn. Syst. 14(3), 597–615 (2006)
Bose, C.J., Murray, R.: Duality and the computation of approximate invariant densities for nonsingular transformations. SIAM J. Optim. 18(2), 691–709 (electronic) (2007). doi:10.1137/060658163
Bruin, H., Demers, M., Melbourne, I.: Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergod. Theor. Dyn. Syst. 30, 687–728 (2010)
Collett, P., MartÃnez, S., Schmitt, B.: The Yorke–Pianigiani measure and the asymptotic law on the limit cantor set of expanding systems. Nonlinearity 7, 1437–1443 (1994)
Collett, P., MartÃnez, S., Schmitt, B.: The Pianigiani–Yorke measure for topological markov chains. Israel J. Math. 97, 61–70 (1997)
Demers, M.: Markov extensions and conditionally invariant measures for certain logistic maps with small holes. Ergod. Theor. Dyn. Syst. 25, 1139–1171 (2005)
Demers, M.: Markov extensions for dynamical systems with holes: an application to expanding maps of the interval. Israel J. Math. 146, 189–221 (2005)
Demers, M.: Dispersing billiards with small holes. In: Ergodic Theory, Open Dynamics, and Coherent Structures. Springer, New York (2014)
Demers, M., Young, L.S.: Escape rates and conditionally invariant measures. Nonlinearity 19, 377–397 (2006). doi:10.1088/0951-7715/19/2/008
Dolgopyat, D., Wright, P.: The diffusion coefficient for piecewise expanding maps of the interval with metastable states. Stoch. Dyn. 12(1), 1150,005, 13 (2012). doi:10.1142/S0219493712003547
Froyland, G.: Extracting dynamical behavior via Markov models. In: Nonlinear Dynamics and Statistics (Cambridge, 1998), pp. 281–321. Birkhäuser, Boston (2001)
Froyland, G., Dellnitz, M.: Detecting and locating near-optimal almost-invariant sets and cycles. SIAM J. Sci. Comput. 24(6), 1839–1863 (electronic) (2003). doi:10.1137/S106482750238911X
Froyland, G., Murray, R., Terhesiu, D.: Efficient computation of topological entropy, pressure, conformal measures, and equilibrium states in one dimension. Phys. Rev. E 76(3), 036702 (2007). doi:10.1103/PhysRevE.76.036702
Froyland, G., Murray, R., Stancevic, O.: Spectral degeneracy and escape dynamics for intermittent maps with a hole. Nonlinearity 24(9), 2435–2463 (2011). doi:10.1088/0951-7715/24/9/003
González-Tokman, C., Hunt, B.R., Wright, P.: Approximating invariant densities of metastable systems. Ergod. Theor. Dyn. Syst. 31(5), 1345–1361 (2011). doi:10.1017/S0143385710000337
Homburg, A.J., Young, T.: Intermittency in families of unimodal maps. Ergod. Theor. Dyn. Syst. 22(1), 203–225 (2002). doi:10.1017/S0143385702000093
Keller, G., Liverani, C.: Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135(3), 519–534 (2009). doi:10.1007/s10955-009-9747-8
Liverani, C., Maume-Deschamps, V.: Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Ann. Inst. H. Poincaré Probab. Stat. 39, 385–412 (2003)
Pianigiani, G., Yorke, J.: Expanding maps on sets which are almost invariant: decay and chaos. Trans. Am. Math. Soc. 252, 351–366 (1979)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Ulam, S.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)
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Bose, C., Murray, R. (2014). Numerical Approximation of Conditionally Invariant Measures via Maximum Entropy. In: Bahsoun, W., Bose, C., Froyland, G. (eds) Ergodic Theory, Open Dynamics, and Coherent Structures. Springer Proceedings in Mathematics & Statistics, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0419-8_5
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