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April  2018, 38(4): 1777-1807. doi: 10.3934/dcds.2018073

## Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus

 1 Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran 2 Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran 3 Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

* Corresponding author: F. H. Ghane

Received  August 2016 Revised  October 2017 Published  January 2018

In this paper we address the existence and ergodicity of non-uniformly hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems have formulation as a skew product system defined by planar diffeomorphisms, with average contraction condition, forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one physical measure. In our approach, these skew product systems arising from iterated function systems which are generated by finitely many weak contractive diffeomorphisms. Under some conditions including negative fiber Lyapunov exponents, we prove the existence of unique non-uniformly hyperbolic attracting invariant graphs for these systems which attract positive orbits of almost all initial points. Also, we prove that these systems are Bernoulli and therefore they are mixing. Moreover, these properties remain true under small perturbations in the space of endomorphisms on the solid torus.

Citation: Marzie Zaj, Abbas Fakhari, Fatemeh Helen Ghane, Azam Ehsani. Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1777-1807. doi: 10.3934/dcds.2018073
##### References:
 [1] C. Åkerlund-Biström, A generalization of Hutchinson distance and applications, Random Comput. Dynam., 5 (1997), 159-176. [2] J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057. [3] A. Arbieto, A. Junqueira and B. Santiago, On weakly hyperbolic iterated function systems, Bulletin of the Brazilian Mathematical Society, New Series, 48 (2017), 111–140, arXiv: 1211.1738v1. doi: 10.1007/s00574-016-0018-4. [4] L. Arnold, Random Dynamical Systems, Springer Verlag, 1998. [5] L. Arnold and H. Crauel, Iterated function systems and multiplicative ergodic theory, in Diffusion Processes and Related Problems in Analysis, Vol. Ⅱ, pages 283–305, Progr. Probab., 27, Birkhauser Boston, (1992). [6] M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo, Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities, Ann. Inst. H. Poincaré Probab. Statist., 24 (1998), 367-394. [7] M. F. Barnsley, J. H. Elton and D. P. Hardin, Recurrent iterated function systems, Constr. Approx, 5 (1989), 3-31. doi: 10.1007/BF01889596. [8] M. F. Barnsley and K. Leśniak, The chaos game on a general iterated function system from a topological point of view, International Journal of Bifurcation and Chaos, 24 (2014), 1450139, 10pp. [9] M. F. Barnsley, K. Lesniak and A. Vince, Symbolic iterated function systems, fast basins and fractal manifolds, arXiv: 1308.3819v3, (2014). [10] P. Berger, S. Crovisier and E. Pujals, Iterated functions systems, blenders and parablenders, Conference of Fractals and Related Fields, Recent Developments in Fractals and Related Fields, (2017), 57-70. [11] A. Bielecki, Iterated function systems analogues on compact metric spaces and their attractors, Univ. Iagel. Acta Math, 32 (1995), 187-192. [12] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central directions is mostly contracting, Israel J. of Math., 115 (2000), 157-193. doi: 10.1007/BF02810585. [13] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Math., 470, Springer Verlag, 1975. [14] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848. [15] D. Broomhead, D. Hadjiloucas and M. Nicol, Random and deterministic perturbation of a class of skew-product systems, Dynamics and Stability of Systems, 14 (1999), 115-128. doi: 10.1080/026811199282029. [16] Y. Bugeaud, Distribution Modulo one and Diophantine Approximation, Cambridge Tracts in Mathematics, 193, Cambridge: Cambridge University Press, 2012. [17] K. M. Campbell, Observational noise in skew product systems, Physica D, 107 (1997), 43-56. doi: 10.1016/S0167-2789(97)00056-0. [18] K. M. Campbell and M. E. Davies, The existence of inertial functions in skew product systems, Nonlinearity, 9 (1996), 801-817. doi: 10.1088/0951-7715/9/3/010. [19] M. Carvalho, Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993), 21-44. [20] N. D. Cong, D. T. Son and S. Siegmund, A computational ergodic theorem for infinite iterated function systems, Stoch. Dyn., 8 (2008), 365-381. doi: 10.1142/S0219493708002354. [21] H. Crauel, Extremal exponents of random dynamical systems do not vanish, J. Dynam. Differential Equations, 2 (1990), 487-499. doi: 10.1007/BF01048947. [22] M. E. Davies and K. M. Campbell, Linear recursive filters and nonlinear dynamics, Nonlinearity, 9 (1996), 487-499. doi: 10.1088/0951-7715/9/2/012. [23] A. Edalat, Power domains and iterated function systems, Inform. and Comput, 124 (1996), 182-197. doi: 10.1006/inco.1996.0014. [24] J. H. Elton, An ergodic theorem for iterated maps, Ergodic Th. and Dynam. sys, 7 (1987), 481-488. [25] J. H. Elton, A multiplicative ergodic theorem for Lipschitz maps, Stochastic Processes and their Applications, 34 (1990), 39-47. doi: 10.1016/0304-4149(90)90055-W. [26] F. Filip and J. Sustek, An elementary proof that almost all real numbers are normal, Acta Univ. Sapientiae, Mathematica, 2 (2010), 99-110. [27] G. Froyland, On the Estimation of Invariant Measures and Lyapunov Exponents Arising from Iid Compositions of Maps, Technical report, 1998. [28] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Stat., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. [29] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Bull. Amer. Math. Soc., 75 (1969), 149-152. doi: 10.1090/S0002-9904-1969-12184-1. [30] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer, 1977. [31] A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory and Dynamical Systems, 32 (2012), 2011-2024. doi: 10.1017/S014338571100068X. [32] B. R. Hunt, E. Ott and J. A. Yorke, Fractal dimensions of chaotic saddles of dynamical systems, Phys. Rev. E, 54 (1996), 4819-4823. doi: 10.1103/PhysRevE.54.4819. [33] B. R. Hunt, E. Ott and J. A. Yorke, Differentiable generalized synchronization of chaos, Phys. Rev. E, 55 (1997), 4029-4034. doi: 10.1103/PhysRevE.55.4029. [34] Y. Ilyashenko, V. Kleptsyn and P. Saltykov, Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins, Journal of Fixed Point Theory and Applications, 3 (2008), 449-463. doi: 10.1007/s11784-008-0088-z. [35] Y. Ilyashenko and A. Negut, Holder properties of perturbed skew products and Fubini regained, Nonlinearity, 25 (2012), 2377-2399. doi: 10.1088/0951-7715/25/8/2377. [36] J. R. Jachymski, An fixed point criterion for continuous self mappings on a complete metric space, Aequations Math., 48 (1994), 163-170. doi: 10.1007/BF01832983. [37] G. Keller, Equilibrium States in Ergodic Theory, LMSST 42 Cambridge University Press, 1998. [38] V. Kleptsyn and D. Volk, Physical measures for random walks on interval, Moscow Mathematical Journal, 14 (2014), 339-365. [39] A. S. Kravchenko, Completeness of the spaces of separable measures in the KantrovichRubinshteǐn metric, (Russian summary) Sibrisk. Math. Zh., 47 (2006), 85–96; translation in Siberian Math. J., 47 (2006), 68–76. [40] U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics 6, Walter de Gruyter, 1985. [41] Yu. G. Kudryashov, Bony attractors, Funkts. Anal. Prilozhen, 44 (2010), 73-76; English transl. [42] Yu. Kudryashov, Des Orbites Périodiques et Des Attracteurs des Systémes Dynamiques, Ph. D. Thesis, École Normale Supérieure de Lyon, Lyons, 2010. [43] R. Mane, Ergodic Theory and Differentiable Dynamics, 8 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1987. [44] J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 485-507. doi: 10.1016/j.anihpc.2005.01.001. [45] L. M. Pecora and T. L. Carroll, Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data, Chaos, 6 (1996), 432-439. doi: 10.1063/1.166186. [46] A. N. Quas, A $C^1$ expanding map of the circle which is not weak-mixing, Israel Journal of Mathematics, 93 (1996), 359-372. doi: 10.1007/BF02761112. [47] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. [48] Y. Sinai, Gibbs measure in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64. [49] J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179. doi: 10.1016/S0167-2789(97)00167-X. [50] J. Stark, Regularity of invariant graphs for forced systems, Ergodic theory and Dynamcal Systems, 19 (1999), 155-199. doi: 10.1017/S0143385799126555. [51] J. Stark and M. E. Davies, Recursive filters driven by chaotic signals, IEE Colloquium on Exploiting Chaos in Signal Processing. IEE Digest, 143 (1994), 1-516. [52] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, 68, Springer, Berlin, 1988. [53] M. Tsujii, Physical measures for partially hyperbolic surface endomorphism, Acta Mathematica, 194 (2005), 37-132. doi: 10.1007/BF02392516. [54] M. Viana and J. Yang, Measure-theoretical properties of center foliations, A chapter in: Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov, 692 (2017), 291–320. [55] D. Volk, Persistent massive attractors of smooth maps, Ergodic theory and Dynamcal Systems, 34 (2014), 693-704. [56] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. [57] P. Walters, An Introduction to Ergodic Theorem, Springer-Verlag, 1982. [58] R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 169-203. [59] L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

show all references

##### References:
 [1] C. Åkerlund-Biström, A generalization of Hutchinson distance and applications, Random Comput. Dynam., 5 (1997), 159-176. [2] J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057. [3] A. Arbieto, A. Junqueira and B. Santiago, On weakly hyperbolic iterated function systems, Bulletin of the Brazilian Mathematical Society, New Series, 48 (2017), 111–140, arXiv: 1211.1738v1. doi: 10.1007/s00574-016-0018-4. [4] L. Arnold, Random Dynamical Systems, Springer Verlag, 1998. [5] L. Arnold and H. Crauel, Iterated function systems and multiplicative ergodic theory, in Diffusion Processes and Related Problems in Analysis, Vol. Ⅱ, pages 283–305, Progr. Probab., 27, Birkhauser Boston, (1992). [6] M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo, Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities, Ann. Inst. H. Poincaré Probab. Statist., 24 (1998), 367-394. [7] M. F. Barnsley, J. H. Elton and D. P. Hardin, Recurrent iterated function systems, Constr. Approx, 5 (1989), 3-31. doi: 10.1007/BF01889596. [8] M. F. Barnsley and K. Leśniak, The chaos game on a general iterated function system from a topological point of view, International Journal of Bifurcation and Chaos, 24 (2014), 1450139, 10pp. [9] M. F. Barnsley, K. Lesniak and A. Vince, Symbolic iterated function systems, fast basins and fractal manifolds, arXiv: 1308.3819v3, (2014). [10] P. Berger, S. Crovisier and E. Pujals, Iterated functions systems, blenders and parablenders, Conference of Fractals and Related Fields, Recent Developments in Fractals and Related Fields, (2017), 57-70. [11] A. Bielecki, Iterated function systems analogues on compact metric spaces and their attractors, Univ. Iagel. Acta Math, 32 (1995), 187-192. [12] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central directions is mostly contracting, Israel J. of Math., 115 (2000), 157-193. doi: 10.1007/BF02810585. [13] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Math., 470, Springer Verlag, 1975. [14] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848. [15] D. Broomhead, D. Hadjiloucas and M. Nicol, Random and deterministic perturbation of a class of skew-product systems, Dynamics and Stability of Systems, 14 (1999), 115-128. doi: 10.1080/026811199282029. [16] Y. Bugeaud, Distribution Modulo one and Diophantine Approximation, Cambridge Tracts in Mathematics, 193, Cambridge: Cambridge University Press, 2012. [17] K. M. Campbell, Observational noise in skew product systems, Physica D, 107 (1997), 43-56. doi: 10.1016/S0167-2789(97)00056-0. [18] K. M. Campbell and M. E. Davies, The existence of inertial functions in skew product systems, Nonlinearity, 9 (1996), 801-817. doi: 10.1088/0951-7715/9/3/010. [19] M. Carvalho, Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993), 21-44. [20] N. D. Cong, D. T. Son and S. Siegmund, A computational ergodic theorem for infinite iterated function systems, Stoch. Dyn., 8 (2008), 365-381. doi: 10.1142/S0219493708002354. [21] H. Crauel, Extremal exponents of random dynamical systems do not vanish, J. Dynam. Differential Equations, 2 (1990), 487-499. doi: 10.1007/BF01048947. [22] M. E. Davies and K. M. Campbell, Linear recursive filters and nonlinear dynamics, Nonlinearity, 9 (1996), 487-499. doi: 10.1088/0951-7715/9/2/012. [23] A. Edalat, Power domains and iterated function systems, Inform. and Comput, 124 (1996), 182-197. doi: 10.1006/inco.1996.0014. [24] J. H. Elton, An ergodic theorem for iterated maps, Ergodic Th. and Dynam. sys, 7 (1987), 481-488. [25] J. H. Elton, A multiplicative ergodic theorem for Lipschitz maps, Stochastic Processes and their Applications, 34 (1990), 39-47. doi: 10.1016/0304-4149(90)90055-W. [26] F. Filip and J. Sustek, An elementary proof that almost all real numbers are normal, Acta Univ. Sapientiae, Mathematica, 2 (2010), 99-110. [27] G. Froyland, On the Estimation of Invariant Measures and Lyapunov Exponents Arising from Iid Compositions of Maps, Technical report, 1998. [28] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Stat., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. [29] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Bull. Amer. Math. Soc., 75 (1969), 149-152. doi: 10.1090/S0002-9904-1969-12184-1. [30] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer, 1977. [31] A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory and Dynamical Systems, 32 (2012), 2011-2024. doi: 10.1017/S014338571100068X. [32] B. R. Hunt, E. Ott and J. A. Yorke, Fractal dimensions of chaotic saddles of dynamical systems, Phys. Rev. E, 54 (1996), 4819-4823. doi: 10.1103/PhysRevE.54.4819. [33] B. R. Hunt, E. Ott and J. A. Yorke, Differentiable generalized synchronization of chaos, Phys. Rev. E, 55 (1997), 4029-4034. doi: 10.1103/PhysRevE.55.4029. [34] Y. Ilyashenko, V. Kleptsyn and P. Saltykov, Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins, Journal of Fixed Point Theory and Applications, 3 (2008), 449-463. doi: 10.1007/s11784-008-0088-z. [35] Y. Ilyashenko and A. Negut, Holder properties of perturbed skew products and Fubini regained, Nonlinearity, 25 (2012), 2377-2399. doi: 10.1088/0951-7715/25/8/2377. [36] J. R. Jachymski, An fixed point criterion for continuous self mappings on a complete metric space, Aequations Math., 48 (1994), 163-170. doi: 10.1007/BF01832983. [37] G. Keller, Equilibrium States in Ergodic Theory, LMSST 42 Cambridge University Press, 1998. [38] V. Kleptsyn and D. Volk, Physical measures for random walks on interval, Moscow Mathematical Journal, 14 (2014), 339-365. [39] A. S. Kravchenko, Completeness of the spaces of separable measures in the KantrovichRubinshteǐn metric, (Russian summary) Sibrisk. Math. Zh., 47 (2006), 85–96; translation in Siberian Math. J., 47 (2006), 68–76. [40] U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics 6, Walter de Gruyter, 1985. [41] Yu. G. Kudryashov, Bony attractors, Funkts. Anal. Prilozhen, 44 (2010), 73-76; English transl. [42] Yu. Kudryashov, Des Orbites Périodiques et Des Attracteurs des Systémes Dynamiques, Ph. D. Thesis, École Normale Supérieure de Lyon, Lyons, 2010. [43] R. Mane, Ergodic Theory and Differentiable Dynamics, 8 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1987. [44] J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 485-507. doi: 10.1016/j.anihpc.2005.01.001. [45] L. M. Pecora and T. L. Carroll, Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data, Chaos, 6 (1996), 432-439. doi: 10.1063/1.166186. [46] A. N. Quas, A $C^1$ expanding map of the circle which is not weak-mixing, Israel Journal of Mathematics, 93 (1996), 359-372. doi: 10.1007/BF02761112. [47] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. [48] Y. Sinai, Gibbs measure in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64. [49] J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179. doi: 10.1016/S0167-2789(97)00167-X. [50] J. Stark, Regularity of invariant graphs for forced systems, Ergodic theory and Dynamcal Systems, 19 (1999), 155-199. doi: 10.1017/S0143385799126555. [51] J. Stark and M. E. Davies, Recursive filters driven by chaotic signals, IEE Colloquium on Exploiting Chaos in Signal Processing. IEE Digest, 143 (1994), 1-516. [52] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, 68, Springer, Berlin, 1988. [53] M. Tsujii, Physical measures for partially hyperbolic surface endomorphism, Acta Mathematica, 194 (2005), 37-132. doi: 10.1007/BF02392516. [54] M. Viana and J. Yang, Measure-theoretical properties of center foliations, A chapter in: Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov, 692 (2017), 291–320. [55] D. Volk, Persistent massive attractors of smooth maps, Ergodic theory and Dynamcal Systems, 34 (2014), 693-704. [56] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. [57] P. Walters, An Introduction to Ergodic Theorem, Springer-Verlag, 1982. [58] R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 169-203. [59] L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717.
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