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May  2020, 40(5): 2903-2915. doi: 10.3934/dcds.2020154

Statistical stability for Barge-Martin attractors derived from tent maps

Philip Boyland 1, , André de Carvalho 2, and  Toby Hall 3,

1. 

Department of Mathematics, University of Florida, 372 Little Hall, Gainesville, FL 32611-8105, USA
2. 

Departamento de Matemática Aplicada, IME-USP, Rua Do Matão 1010, Cidade Universitária, 05508-090 São Paulo SP, Brazil
3. 

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK

Received  August 2019 Revised  November 2019 Published  March 2020

Fund Project: The authors are grateful for the support of FAPESP grant 2016/25053-8 and CAPES grant 88881.119100/2016-01.

Let $ \{f_t\}_{t\in(1,2]} $ be the family of core tent maps of slopes $ t $. The parameterized Barge-Martin construction yields a family of disk homeomorphisms $ \Phi_t\colon D^2\to D^2 $, having transitive global attractors $ \Lambda_t $ on which $ \Phi_t $ is topologically conjugate to the natural extension of $ f_t $. The unique family of absolutely continuous invariant measures for $ f_t $ induces a family of ergodic $ \Phi_t $-invariant measures $ \nu_t $, supported on the attractors $ \Lambda_t $.

We show that this family $ \nu_t $ varies weakly continuously, and that the measures $ \nu_t $ are physical with respect to a weakly continuously varying family of background Oxtoby-Ulam measures $ \rho_t $.

Similar results are obtained for the family $ \chi_t\colon S^2\to S^2 $ of transitive sphere homeomorphisms, constructed in a previous paper of the authors as factors of the natural extensions of $ f_t $.

Keywords: Statistical stability, inverse limits, attractors, tent maps.
Mathematics Subject Classification: 37A10, 37B45, 37C75, 37E05.
Citation: Philip Boyland, André de Carvalho, Toby Hall. Statistical stability for Barge-Martin attractors derived from tent maps. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2903-2915. doi: 10.3934/dcds.2020154
References:
[1] S. Alpern and V. Prasad, Typical Dynamics of Volume Preserving Homeomorphisms, Cambridge Tracts in Mathematics, 139. Cambridge University Press, Cambridge, 2000.   Google Scholar
[2]

J. F. Alves, Strong statistical stability of non-uniformly expanding maps, Nonlinearity, 17 (2004), 1193-1215.  doi: 10.1088/0951-7715/17/4/004.  Google Scholar

[3]

J. F. Alves, M. Carvalho and J. Freitas, Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures, Comm. Math. Phys., 296 (2010), 739-767.  doi: 10.1007/s00220-010-1027-6.  Google Scholar

[4]

J. F. Alves, M. Carvalho and J. Freitas, Statistical stability for Hénon maps of the Benedicks-Carleson type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 595-637.  doi: 10.1016/j.anihpc.2009.09.009.  Google Scholar

[5]

J. Alves and A. Pumariño, Entropy formula and continuity of entropy for piecewise expanding maps, arXiv: 1806.01095 [math.DS]. Google Scholar

[6]

J. F. Alves, A. Pumariño and E. Vigil, Statistical stability for multidimensional piecewise expanding maps, Proc. Amer. Math. Soc., 145 (2017), 3057-3068.  doi: 10.1090/proc/13518.  Google Scholar

[7]

J. F. Alves and M. Soufi, Statistical stability and limit laws for Rovella maps, Nonlinearity, 25 (2012), 3527-3552.  doi: 10.1088/0951-7715/25/12/3527.  Google Scholar

[8]

J. F. Alves and M. Soufi, Statistical stability in chaotic dynamics, Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., Springer, Heidelberg, 54 (2013), 7-24.  doi: 10.1007/978-3-642-38830-9_2.  Google Scholar

[9]

J. F. Alves and M. Soufi, Statistical stability of geometric Lorenz attractors, Fund. Math., 224 (2014), 219-231.  doi: 10.4064/fm224-3-2.  Google Scholar

[10]

J. F. Alves and M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems, 22 (2002), 1-32.  doi: 10.1017/S0143385702000019.  Google Scholar

[11]

M. Barge, Prime end rotation numbers associated with the Hénon maps, Continuum Theory and Dynamical Systems, Lecture Notes in Pure and Appl. Math., Dekker, New York, 149 (1993), 15-33.   Google Scholar

[12]

M. Barge, K. Brucks and B. Diamond, Self-similarity in inverse limit spaces of the tent family, Proc. Amer. Math. Soc., 124 (1996), 3563-3570.  doi: 10.1090/S0002-9939-96-03690-8.  Google Scholar

[13]

M. Barge, H. Bruin and S. Štimac, The Ingram conjecture, Geom. Topol., 16 (2012), 2481-2516.  doi: 10.2140/gt.2012.16.2481.  Google Scholar

[14]

M. Barge and S. Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity, 8 (1995), 29-42.  doi: 10.1088/0951-7715/8/1/003.  Google Scholar

[15]

M. Barge and J. Martin, The construction of global attractors, Proc. Amer. Math. Soc., 110 (1990), 523-525.  doi: 10.1090/S0002-9939-1990-1023342-1.  Google Scholar

[16]

M. Benedicks and L.-S. Young, Sinaǐ-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576.  doi: 10.1007/BF01232446.  Google Scholar

[17]

P. Billingsley, Convergence of Probability Measures, Second edition. Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[18]

P. Boyland, A. de Carvalho and T. Hall, Inverse limits as attractors in parameterized families, Bull. London Math. Soc., 45 (2013), 1075-1085.  doi: 10.1112/blms/bdt032.  Google Scholar

[19]

P. Boyland, A. de Carvalho and T. Hall, Natural extensions of unimodal maps: Virtual sphere homeomorphisms and prime ends of basin boundaries, arXiv: 1704.06624v3 [math.DS]. Google Scholar

[20]

M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc., 11 (1960), 478-483.  doi: 10.1090/S0002-9939-1960-0115157-4.  Google Scholar

[21]

H. Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl., 96 (1999), 191-208.  doi: 10.1016/S0166-8641(98)00054-6.  Google Scholar

[22]

A. Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup.(4), 13 (1980), 45–93. doi: 10.24033/asens.1377.  Google Scholar

[23]

J. Kennedy, B. E. Raines and D. R. Stockman, Basins of measures on inverse limit spaces for the induced homeomorphism, Ergodic Theory Dynam. Systems, 30 (2010), 1119-1130.  doi: 10.1017/S0143385709000388.  Google Scholar

[24]

J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math.(2), 42 (1941), 874-920.  doi: 10.2307/1968772.  Google Scholar

[25]

B. E. Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math., 182 (2004), 241-268.  doi: 10.4064/fm182-3-4.  Google Scholar

[26]

C. H. Vásquez, Statistical stability for diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems, 27 (2007), 253-283.  doi: 10.1017/S0143385706000721.  Google Scholar

[27]

M. Viana, Lecture Notes on Attractors and Physical Measures, Monografís del Instituto de Matemática y Ciencias Afines, 8. Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999.  Google Scholar

[28]

Q. D. Wang and L.-S. Young, Strange attractors with one direction of instability, Comm. Math. Phys., 218 (2001), 1-97.  doi: 10.1007/s002200100379.  Google Scholar

[29]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.  Google Scholar

show all references

References:
[1] S. Alpern and V. Prasad, Typical Dynamics of Volume Preserving Homeomorphisms, Cambridge Tracts in Mathematics, 139. Cambridge University Press, Cambridge, 2000.   Google Scholar
[2]

J. F. Alves, Strong statistical stability of non-uniformly expanding maps, Nonlinearity, 17 (2004), 1193-1215.  doi: 10.1088/0951-7715/17/4/004.  Google Scholar

[3]

J. F. Alves, M. Carvalho and J. Freitas, Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures, Comm. Math. Phys., 296 (2010), 739-767.  doi: 10.1007/s00220-010-1027-6.  Google Scholar

[4]

J. F. Alves, M. Carvalho and J. Freitas, Statistical stability for Hénon maps of the Benedicks-Carleson type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 595-637.  doi: 10.1016/j.anihpc.2009.09.009.  Google Scholar

[5]

J. Alves and A. Pumariño, Entropy formula and continuity of entropy for piecewise expanding maps, arXiv: 1806.01095 [math.DS]. Google Scholar

[6]

J. F. Alves, A. Pumariño and E. Vigil, Statistical stability for multidimensional piecewise expanding maps, Proc. Amer. Math. Soc., 145 (2017), 3057-3068.  doi: 10.1090/proc/13518.  Google Scholar

[7]

J. F. Alves and M. Soufi, Statistical stability and limit laws for Rovella maps, Nonlinearity, 25 (2012), 3527-3552.  doi: 10.1088/0951-7715/25/12/3527.  Google Scholar

[8]

J. F. Alves and M. Soufi, Statistical stability in chaotic dynamics, Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., Springer, Heidelberg, 54 (2013), 7-24.  doi: 10.1007/978-3-642-38830-9_2.  Google Scholar

[9]

J. F. Alves and M. Soufi, Statistical stability of geometric Lorenz attractors, Fund. Math., 224 (2014), 219-231.  doi: 10.4064/fm224-3-2.  Google Scholar

[10]

J. F. Alves and M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems, 22 (2002), 1-32.  doi: 10.1017/S0143385702000019.  Google Scholar

[11]

M. Barge, Prime end rotation numbers associated with the Hénon maps, Continuum Theory and Dynamical Systems, Lecture Notes in Pure and Appl. Math., Dekker, New York, 149 (1993), 15-33.   Google Scholar

[12]

M. Barge, K. Brucks and B. Diamond, Self-similarity in inverse limit spaces of the tent family, Proc. Amer. Math. Soc., 124 (1996), 3563-3570.  doi: 10.1090/S0002-9939-96-03690-8.  Google Scholar

[13]

M. Barge, H. Bruin and S. Štimac, The Ingram conjecture, Geom. Topol., 16 (2012), 2481-2516.  doi: 10.2140/gt.2012.16.2481.  Google Scholar

[14]

M. Barge and S. Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity, 8 (1995), 29-42.  doi: 10.1088/0951-7715/8/1/003.  Google Scholar

[15]

M. Barge and J. Martin, The construction of global attractors, Proc. Amer. Math. Soc., 110 (1990), 523-525.  doi: 10.1090/S0002-9939-1990-1023342-1.  Google Scholar

[16]

M. Benedicks and L.-S. Young, Sinaǐ-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576.  doi: 10.1007/BF01232446.  Google Scholar

[17]

P. Billingsley, Convergence of Probability Measures, Second edition. Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[18]

P. Boyland, A. de Carvalho and T. Hall, Inverse limits as attractors in parameterized families, Bull. London Math. Soc., 45 (2013), 1075-1085.  doi: 10.1112/blms/bdt032.  Google Scholar

[19]

P. Boyland, A. de Carvalho and T. Hall, Natural extensions of unimodal maps: Virtual sphere homeomorphisms and prime ends of basin boundaries, arXiv: 1704.06624v3 [math.DS]. Google Scholar

[20]

M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc., 11 (1960), 478-483.  doi: 10.1090/S0002-9939-1960-0115157-4.  Google Scholar

[21]

H. Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl., 96 (1999), 191-208.  doi: 10.1016/S0166-8641(98)00054-6.  Google Scholar

[22]

A. Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup.(4), 13 (1980), 45–93. doi: 10.24033/asens.1377.  Google Scholar

[23]

J. Kennedy, B. E. Raines and D. R. Stockman, Basins of measures on inverse limit spaces for the induced homeomorphism, Ergodic Theory Dynam. Systems, 30 (2010), 1119-1130.  doi: 10.1017/S0143385709000388.  Google Scholar

[24]

J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math.(2), 42 (1941), 874-920.  doi: 10.2307/1968772.  Google Scholar

[25]

B. E. Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math., 182 (2004), 241-268.  doi: 10.4064/fm182-3-4.  Google Scholar

[26]

C. H. Vásquez, Statistical stability for diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems, 27 (2007), 253-283.  doi: 10.1017/S0143385706000721.  Google Scholar

[27]

M. Viana, Lecture Notes on Attractors and Physical Measures, Monografís del Instituto de Matemática y Ciencias Afines, 8. Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999.  Google Scholar

[28]

Q. D. Wang and L.-S. Young, Strange attractors with one direction of instability, Comm. Math. Phys., 218 (2001), 1-97.  doi: 10.1007/s002200100379.  Google Scholar

[29]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.  Google Scholar

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