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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. Google Scholar

[2]

J. F. AlvesC. 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. Google Scholar

[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. Google Scholar

[4]

L. Arnold, Random Dynamical Systems, Springer Verlag, 1998. Google Scholar

[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). Google Scholar

[6]

M. F. BarnsleyS. G. DemkoJ. 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. Google Scholar

[7]

M. F. BarnsleyJ. H. Elton and D. P. Hardin, Recurrent iterated function systems, Constr. Approx, 5 (1989), 3-31. doi: 10.1007/BF01889596. Google Scholar

[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. Google Scholar

[9]

M. F. Barnsley, K. Lesniak and A. Vince, Symbolic iterated function systems, fast basins and fractal manifolds, arXiv: 1308.3819v3, (2014).Google Scholar

[10]

P. BergerS. 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. Google Scholar

[11]

A. Bielecki, Iterated function systems analogues on compact metric spaces and their attractors, Univ. Iagel. Acta Math, 32 (1995), 187-192. Google Scholar

[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. Google Scholar

[13]

R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Math., 470, Springer Verlag, 1975. Google Scholar

[14]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848. Google Scholar

[15]

D. BroomheadD. 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. Google Scholar

[16]

Y. Bugeaud, Distribution Modulo one and Diophantine Approximation, Cambridge Tracts in Mathematics, 193, Cambridge: Cambridge University Press, 2012. Google Scholar

[17]

K. M. Campbell, Observational noise in skew product systems, Physica D, 107 (1997), 43-56. doi: 10.1016/S0167-2789(97)00056-0. Google Scholar

[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. Google Scholar

[19]

M. Carvalho, Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993), 21-44. Google Scholar

[20]

N. D. CongD. T. Son and S. Siegmund, A computational ergodic theorem for infinite iterated function systems, Stoch. Dyn., 8 (2008), 365-381. doi: 10.1142/S0219493708002354. Google Scholar

[21]

H. Crauel, Extremal exponents of random dynamical systems do not vanish, J. Dynam. Differential Equations, 2 (1990), 487-499. doi: 10.1007/BF01048947. Google Scholar

[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. Google Scholar

[23]

A. Edalat, Power domains and iterated function systems, Inform. and Comput, 124 (1996), 182-197. doi: 10.1006/inco.1996.0014. Google Scholar

[24]

J. H. Elton, An ergodic theorem for iterated maps, Ergodic Th. and Dynam. sys, 7 (1987), 481-488. Google Scholar

[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. Google Scholar

[26]

F. Filip and J. Sustek, An elementary proof that almost all real numbers are normal, Acta Univ. Sapientiae, Mathematica, 2 (2010), 99-110. Google Scholar

[27]

G. Froyland, On the Estimation of Invariant Measures and Lyapunov Exponents Arising from Iid Compositions of Maps, Technical report, 1998.Google Scholar

[28]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Stat., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. Google Scholar

[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. Google Scholar

[30]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer, 1977. Google Scholar

[31]

A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory and Dynamical Systems, 32 (2012), 2011-2024. doi: 10.1017/S014338571100068X. Google Scholar

[32]

B. R. HuntE. 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. Google Scholar

[33]

B. R. HuntE. Ott and J. A. Yorke, Differentiable generalized synchronization of chaos, Phys. Rev. E, 55 (1997), 4029-4034. doi: 10.1103/PhysRevE.55.4029. Google Scholar

[34]

Y. IlyashenkoV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

G. Keller, Equilibrium States in Ergodic Theory, LMSST 42 Cambridge University Press, 1998. Google Scholar

[38]

V. Kleptsyn and D. Volk, Physical measures for random walks on interval, Moscow Mathematical Journal, 14 (2014), 339-365. Google Scholar

[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. Google Scholar

[40]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics 6, Walter de Gruyter, 1985. Google Scholar

[41]

Yu. G. Kudryashov, Bony attractors, Funkts. Anal. Prilozhen, 44 (2010), 73-76; English transl. Google Scholar

[42]

Yu. Kudryashov, Des Orbites Périodiques et Des Attracteurs des Systémes Dynamiques, Ph. D. Thesis, École Normale Supérieure de Lyon, Lyons, 2010.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[47]

D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. Google Scholar

[48]

Y. Sinai, Gibbs measure in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64. Google Scholar

[49]

J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179. doi: 10.1016/S0167-2789(97)00167-X. Google Scholar

[50]

J. Stark, Regularity of invariant graphs for forced systems, Ergodic theory and Dynamcal Systems, 19 (1999), 155-199. doi: 10.1017/S0143385799126555. Google Scholar

[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. Google Scholar

[52]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, 68, Springer, Berlin, 1988. Google Scholar

[53]

M. Tsujii, Physical measures for partially hyperbolic surface endomorphism, Acta Mathematica, 194 (2005), 37-132. doi: 10.1007/BF02392516. Google Scholar

[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. Google Scholar

[55]

D. Volk, Persistent massive attractors of smooth maps, Ergodic theory and Dynamcal Systems, 34 (2014), 693-704. Google Scholar

[56]

P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. Google Scholar

[57]

P. Walters, An Introduction to Ergodic Theorem, Springer-Verlag, 1982. Google Scholar

[58]

R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 169-203. Google Scholar

[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. Google Scholar

show all references

References:
[1]

C. Åkerlund-Biström, A generalization of Hutchinson distance and applications, Random Comput. Dynam., 5 (1997), 159-176. Google Scholar

[2]

J. F. AlvesC. 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. Google Scholar

[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. Google Scholar

[4]

L. Arnold, Random Dynamical Systems, Springer Verlag, 1998. Google Scholar

[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). Google Scholar

[6]

M. F. BarnsleyS. G. DemkoJ. 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. Google Scholar

[7]

M. F. BarnsleyJ. H. Elton and D. P. Hardin, Recurrent iterated function systems, Constr. Approx, 5 (1989), 3-31. doi: 10.1007/BF01889596. Google Scholar

[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. Google Scholar

[9]

M. F. Barnsley, K. Lesniak and A. Vince, Symbolic iterated function systems, fast basins and fractal manifolds, arXiv: 1308.3819v3, (2014).Google Scholar

[10]

P. BergerS. 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. Google Scholar

[11]

A. Bielecki, Iterated function systems analogues on compact metric spaces and their attractors, Univ. Iagel. Acta Math, 32 (1995), 187-192. Google Scholar

[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. Google Scholar

[13]

R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Math., 470, Springer Verlag, 1975. Google Scholar

[14]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848. Google Scholar

[15]

D. BroomheadD. 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. Google Scholar

[16]

Y. Bugeaud, Distribution Modulo one and Diophantine Approximation, Cambridge Tracts in Mathematics, 193, Cambridge: Cambridge University Press, 2012. Google Scholar

[17]

K. M. Campbell, Observational noise in skew product systems, Physica D, 107 (1997), 43-56. doi: 10.1016/S0167-2789(97)00056-0. Google Scholar

[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. Google Scholar

[19]

M. Carvalho, Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993), 21-44. Google Scholar

[20]

N. D. CongD. T. Son and S. Siegmund, A computational ergodic theorem for infinite iterated function systems, Stoch. Dyn., 8 (2008), 365-381. doi: 10.1142/S0219493708002354. Google Scholar

[21]

H. Crauel, Extremal exponents of random dynamical systems do not vanish, J. Dynam. Differential Equations, 2 (1990), 487-499. doi: 10.1007/BF01048947. Google Scholar

[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. Google Scholar

[23]

A. Edalat, Power domains and iterated function systems, Inform. and Comput, 124 (1996), 182-197. doi: 10.1006/inco.1996.0014. Google Scholar

[24]

J. H. Elton, An ergodic theorem for iterated maps, Ergodic Th. and Dynam. sys, 7 (1987), 481-488. Google Scholar

[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. Google Scholar

[26]

F. Filip and J. Sustek, An elementary proof that almost all real numbers are normal, Acta Univ. Sapientiae, Mathematica, 2 (2010), 99-110. Google Scholar

[27]

G. Froyland, On the Estimation of Invariant Measures and Lyapunov Exponents Arising from Iid Compositions of Maps, Technical report, 1998.Google Scholar

[28]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Stat., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. Google Scholar

[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. Google Scholar

[30]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer, 1977. Google Scholar

[31]

A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory and Dynamical Systems, 32 (2012), 2011-2024. doi: 10.1017/S014338571100068X. Google Scholar

[32]

B. R. HuntE. 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. Google Scholar

[33]

B. R. HuntE. Ott and J. A. Yorke, Differentiable generalized synchronization of chaos, Phys. Rev. E, 55 (1997), 4029-4034. doi: 10.1103/PhysRevE.55.4029. Google Scholar

[34]

Y. IlyashenkoV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

G. Keller, Equilibrium States in Ergodic Theory, LMSST 42 Cambridge University Press, 1998. Google Scholar

[38]

V. Kleptsyn and D. Volk, Physical measures for random walks on interval, Moscow Mathematical Journal, 14 (2014), 339-365. Google Scholar

[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. Google Scholar

[40]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics 6, Walter de Gruyter, 1985. Google Scholar

[41]

Yu. G. Kudryashov, Bony attractors, Funkts. Anal. Prilozhen, 44 (2010), 73-76; English transl. Google Scholar

[42]

Yu. Kudryashov, Des Orbites Périodiques et Des Attracteurs des Systémes Dynamiques, Ph. D. Thesis, École Normale Supérieure de Lyon, Lyons, 2010.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[47]

D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. Google Scholar

[48]

Y. Sinai, Gibbs measure in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64. Google Scholar

[49]

J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179. doi: 10.1016/S0167-2789(97)00167-X. Google Scholar

[50]

J. Stark, Regularity of invariant graphs for forced systems, Ergodic theory and Dynamcal Systems, 19 (1999), 155-199. doi: 10.1017/S0143385799126555. Google Scholar

[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. Google Scholar

[52]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, 68, Springer, Berlin, 1988. Google Scholar

[53]

M. Tsujii, Physical measures for partially hyperbolic surface endomorphism, Acta Mathematica, 194 (2005), 37-132. doi: 10.1007/BF02392516. Google Scholar

[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. Google Scholar

[55]

D. Volk, Persistent massive attractors of smooth maps, Ergodic theory and Dynamcal Systems, 34 (2014), 693-704. Google Scholar

[56]

P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. Google Scholar

[57]

P. Walters, An Introduction to Ergodic Theorem, Springer-Verlag, 1982. Google Scholar

[58]

R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 169-203. Google Scholar

[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. Google Scholar

Figure 1.  Constructing a Cusp-like Region
Figure 2.  Blending Region
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