Linear response for intermittent maps with critical point. (English) Zbl 07867457
Summary: We consider a two-parameter family of maps \(T_{\alpha,\beta}:[0,1]\to[0,1]\) with a neutral fixed point and a non-flat critical point. Building on a cone technique due to V. Baladi and M. Todd [Commun. Math. Phys. 347, No. 3, 857–874 (2016; Zbl 1381.37005)], we show that for a class of \(L^q\) observables \(\phi:[0,1]\to\mathbb{R}\) the bivariate map \((\alpha,\beta)\mapsto\int_0^1\phi\mathrm{d}\mu_{\alpha,\beta}\), where \(\mu_{\alpha,\beta}\) denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.
{© 2024 IOP Publishing Ltd & London Mathematical Society}
{© 2024 IOP Publishing Ltd & London Mathematical Society}
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 37A05 | Dynamical aspects of measure-preserving transformations |
Citations:
Zbl 1381.37005References:
| [1] | Aimino, R.; Huyi, H.; Nicol, M.; Török, A.; Vaienti, S., Polynomial loss of memory for maps of the interval with a neutral fixed point, Discrete Contin. Dyn. Syst., 35, 793-806, 2015 · Zbl 1351.37163 · doi:10.3934/dcds.2015.35.793 |
| [2] | Aspenberg, M.; Baladi, V.; Leppänen, J.; Persson, T., On the fractional susceptibility function of piecewise expanding maps, Discrete Contin. Dyn. Syst., 42, 679-706, 2022 · Zbl 1493.37047 · doi:10.3934/dcds.2021133 |
| [3] | Bahsoun, W.; Bose, C.; Duan, Y., Decay of correlation for random intermittent maps, Nonlinearity, 27, 1543-54, 2014 · Zbl 1348.37064 · doi:10.1088/0951-7715/27/7/1543 |
| [4] | Bahsoun, W.; Korepanov, A., Statistical aspects of mean field coupled intermittent maps, Ergod. Theor. Dynam. Syst., 1-13, 2023 · Zbl 07861137 · doi:10.1017/etds.2023.53 |
| [5] | Bahsoun, W.; Ruziboev, M.; Saussol, B., Linear response for random dynamical systems, Adv. Math., 364, 2020 · Zbl 1475.37059 · doi:10.1016/j.aim.2020.107011 |
| [6] | Bahsoun, W.; Saussol, B., Linear response in the intermittent family: differentiation in a weighted C^0-norm, Discrete Contin. Dyn. Syst., 36, 6657-68, 2016 · Zbl 1368.37043 · doi:10.3934/dcds.2016089 |
| [7] | Baladi, V., On the susceptibility function of piecewise expanding interval maps, Commun. Math. Phys., 275, 839-59, 2007 · Zbl 1140.37327 · doi:10.1007/s00220-007-0320-5 |
| [8] | Baladi, V., Linear response, or else, pp 525-45, 2014 · Zbl 1373.37074 |
| [9] | Baladi, V.; Smania, D., Fractional susceptibility functions for the quadratic family: Misiurewicz-Thurston parameters, Commun. Math. Phys., 385, 1957-2007, 2021 · Zbl 1478.37044 · doi:10.1007/s00220-021-04015-z |
| [10] | Baladi, V.; Todd, M., Linear response for intermittent maps, Commun. Math. Phys., 347, 857-74, 2016 · Zbl 1381.37005 · doi:10.1007/s00220-016-2577-z |
| [11] | Bunimovich, L. A.; Yaofeng, S., Maximal large deviations and slow recurrences in weakly chaotic systems, Adv. Math., 432, 2023 · Zbl 1542.37006 · doi:10.1016/j.aim.2023.109267 |
| [12] | Coates, D.; Luzzatto, S.; Muhammad, M., Doubly intermittent full branch maps with critical points and singularities, 2022 |
| [13] | Crimmins, H.; Nakano, Y., A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics, Ergod. Theor. Dynam. Syst., 1-32, 2023 · Zbl 07861140 · doi:10.1017/etds.2023.41 |
| [14] | Cui, H., Invariant densities for intermittent maps with critical points, J. Differ. Equ. Appl., 27, 404-21, 2021 · Zbl 1471.37041 · doi:10.1080/10236198.2021.1900142 |
| [15] | De Melo, W.; Van Strien, S., One-Dimensional Dynamics, vol 25, 2012, Springer |
| [16] | Dolgopyat, D., On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155, 389-449, 2004 · Zbl 1059.37021 · doi:10.1007/s00222-003-0324-5 |
| [17] | Dragičević, D.; Giulietti, P.; Sedro, J., Quenched linear response for smooth expanding on average cocycles, Commun. Math. Phys., 399, 423-52, 2023 · Zbl 1515.37051 · doi:10.1007/s00220-022-04560-1 |
| [18] | Freitas, J. M.; Todd, M., The statistical stability of equilibrium states for interval maps, Nonlinearity, 22, 259-81, 2009 · Zbl 1166.37014 · doi:10.1088/0951-7715/22/2/002 |
| [19] | Galatolo, S.; Giulietti, P., A linear response for dynamical systems with additive noise, Nonlinearity, 32, 2269-301, 2019 · Zbl 1414.37002 · doi:10.1088/1361-6544/ab0c2e |
| [20] | Gouëzel, S., Sharp polynomial estimates for the decay of correlations, Israel J. Math., 139, 29-65, 2004 · Zbl 1070.37003 · doi:10.1007/BF02787541 |
| [21] | Herfurth, T.; Tchumatchenko, T., How linear response shaped models of neural circuits and the quest for alternatives, Curr. Opin. Neurobiol., 46, 234-40, 2017 · doi:10.1016/j.conb.2017.09.001 |
| [22] | Inoue, T., Asymptotic stability of densities for piecewise convex maps, Ann. Polon. Math., 57, 83-90, 1992 · Zbl 0761.28011 · doi:10.4064/ap-57-1-83-90 |
| [23] | Inoue, T., Weakly attracting repellors for piecewise convex maps, Japan J. Indust. Appl. Math., 9, 413-30, 1992 · Zbl 0772.58036 · doi:10.1007/BF03167275 |
| [24] | Inoue, T.; Toyokawa, H., Invariant measures for random piecewise convex maps, 2023 |
| [25] | Koltai, P.; Cheng Lie, H.; Plonka, M., Fréchet differentiable drift dependence of Perron-Frobenius and Koopman operators for non-deterministic dynamics, Nonlinearity, 32, 4232-57, 2019 · Zbl 1475.60104 · doi:10.1088/1361-6544/ab1f2a |
| [26] | Korepanov, A.; Kosloff, Z.; Melbourne, I., Explicit coupling argument for non-uniformly hyperbolic transformations, Proc. R. Soc. A, 149, 101-30, 2019 · Zbl 1430.37026 · doi:10.1017/S0308210518000161 |
| [27] | Korepanov, A., Linear response for intermittent maps with summable and nonsummable decay of correlations, Nonlinearity, 29, 1735-54, 2016 · Zbl 1360.37100 · doi:10.1088/0951-7715/29/6/1735 |
| [28] | Korepanov, A.; Leppänen, J., Loss of memory and moment bounds for nonstationary intermittent dynamical systems, Commun. Math. Phys., 385, 905-35, 2021 · Zbl 1477.37049 · doi:10.1007/s00220-021-04071-5 |
| [29] | Liverani, C.; Saussol, B.; Vaienti, S., A probabilistic approach to intermittency, Ergod. Theor. Dynam. Syst., 19, 671-85, 1999 · Zbl 0988.37035 · doi:10.1017/S0143385799133856 |
| [30] | Lucarini, V.; Sarno, S., A statistical mechanical approach for the computation of the climatic response to general forcings, Nonlinear Process. Geophys., 18, 7-28, 2011 · doi:10.5194/npg-18-7-2011 |
| [31] | Nicol, M.; Perez Pereira, F.; Török, A., Large deviations and central limit theorems for sequential and random systems of intermittent maps, Ergod. Theor. Dynam. Syst., 41, 2805-32, 2021 · Zbl 1482.37010 · doi:10.1017/etds.2020.90 |
| [32] | Nisoli, I.; Taylor-Crush, T., Rigorous computation of linear response for intermittent maps, 2022 |
| [33] | Pomeau, Y.; Manneville, P., Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., 74, 189-97, 1980 · doi:10.1007/BF01197757 |
| [34] | Ragone, F.; Lucarini, V.; Lunkeit, F., A new framework for climate sensitivity and prediction: a modelling perspective, Clim. Dyn., 46, 1459-71, 2016 · doi:10.1007/s00382-015-2657-3 |
| [35] | Ruelle, D., Differentiation of SRB states, Commun. Math. Phys., 187, 227-41, 1997 · Zbl 0895.58045 · doi:10.1007/s002200050134 |
| [36] | Ruelle, D., A review of linear response theory for general differentiable dynamical systems, Nonlinearity, 22, 855-70, 2009 · Zbl 1158.37305 · doi:10.1088/0951-7715/22/4/009 |
| [37] | Ruelle, D., Structure and f-dependence of the A.C.I.M. for a unimodal map f is Misiurewicz type, Commun. Math. Phys., 287, 1039-70, 2009 · Zbl 1202.37008 · doi:10.1007/s00220-008-0637-8 |
| [38] | Sarig, O., Subexponential decay of correlations, Invent. Math., 150, 629-53, 2002 · Zbl 1042.37005 · doi:10.1007/s00222-002-0248-5 |
| [39] | Sedro, J., Pre-threshold fractional susceptibility functions at Misiurewicz parameters, Nonlinearity, 34, 7174-84, 2021 · Zbl 1480.37037 · doi:10.1088/1361-6544/ac20a3 |
| [40] | Sélley, F. M., Differentiability of the diffusion coefficient for a family of intermittent maps, J. Dyn. Control Syst., 29, 787-804, 2022 · Zbl 1542.37026 · doi:10.1007/s10883-022-09617-x |
| [41] | Sélley, F. M.; Tanzi, M., Linear response for a family of self-consistent transfer operators, Commun. Math. Phys., 382, 1601-24, 2021 · Zbl 1465.37051 · doi:10.1007/s00220-021-03983-6 |
| [42] | Yaofeng, S., Vector-valued almost sure invariance principles for (non)stationary and random dynamical systems, Trans. Am. Math. Soc., 375, 4809-48, 2022 · Zbl 1505.37064 · doi:10.1090/tran/8609 |
| [43] | Young, L-S, Recurrence times and rates of mixing, Israel J. Math., 110, 153-88, 1999 · Zbl 0983.37005 · doi:10.1007/BF02808180 |
| [44] | Zhang, Z., On the smooth dependence of SRB measures for partially hyperbolic systems, Commun. Math. Phys., 358, 45-79, 2018 · Zbl 1456.37035 · doi:10.1007/s00220-018-3088-x |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
