October  2020, 40(10): 6015-6041. doi: 10.3934/dcds.2020257

Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems

1. 

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

2. 

Institute for Advanced Study, Princeton, New Jersey 08540, USA

* Corresponding author

Received  March 2020 Revised  May 2020 Published  June 2020

We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.

Citation: Matteo Tanzi, Lai-Sang Young. Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 6015-6041. doi: 10.3934/dcds.2020257
References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Global dominated splittings and the ${C}^1$ Newhouse phenomenon, Proc. Amer. Math. Soc., 134 (2006), 2229-2237.  doi: 10.1090/S0002-9939-06-08445-0.

[2]

V. Baladi, M. Degli Esposti, S. Isola, E. Järvenpää and A. Kupiainen, The spectrum of weakly coupled map lattices, J. Math. Pures Appl. (9), 77 (1998), 539–584. doi: 10.1016/S0021-7824(98)80138-4.

[3]

P. BálintT. GilbertP. NándoriD. Szász and and I. P. Tóth, On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas, J. Stat. Phys., 166 (2017), 903-925.  doi: 10.1007/s10955-016-1598-5.

[4]

J.-B. BardetG. Keller and and R. Zweimüller, Stochastically stable globally coupled maps with bistable thermodynamic limit, Comm. Math. Phys., 292 (2009), 237-270.  doi: 10.1007/s00220-009-0854-9.

[5]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73–169. doi: 10.2307/2944326.

[6]

M. Benedicks and L.-S. Young, Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576.  doi: 10.1007/BF01232446.

[7]

K. Bjerklöv, A note on circle maps driven by strongly expanding endomorphisms on $\Bbb{T}$, Dyn. Syst., 33 (2018), 361-368.  doi: 10.1080/14689367.2017.1386161.

[8]

S. BoccalettiV. LatoraY. MorenoM. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.

[9]

J. Bochi and M. Viana, Lyapunov Exponents: How frequently are dynamical systems hyperbolic?, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004,271–297.

[10]

C. Bonatti, L. J. Díaz and E. R. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2), 158 (2003), 355–418. doi: 10.4007/annals.2003.158.355.

[11]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585.

[12]

C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508.  doi: 10.1016/j.top.2004.10.009.

[13]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/BFb0081279.

[14]

J. Bricmont and A. Kupiainen, High temperature expansions and dynamical systems, Comm. Math. Phys., 178 (1996), 703-732.  doi: 10.1007/BF02108821.

[15]

M. I. Brin and J. B. Pesin, Partially hyperbolic dynamical systems, Uspehi Mat. Nauk, 28 (1973), 169-170. 

[16]

L. BunimovichC. LiveraniA. Pellegrinotti and Y. Suhov, Ergodic systems of $n$ balls in a billiard table, Comm. Math. Phys., 146 (1992), 357-396.  doi: 10.1007/BF02102633.

[17]

L. A. Bunimovich and Y. G. Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1988), 491-516.  doi: 10.1088/0951-7715/1/4/001.

[18]

J.-R. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics, 671, Springer, Berlin, 2005. doi: 10.1007/b103930.

[19]

H. de Jong, Modeling and simulation of genetic regulatory systems: A literature review, J. Comput. Biol., 9 (2002), 67-103.  doi: 10.1089/10665270252833208.

[20]

L. J. DíazE. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.  doi: 10.1007/BF02392945.

[21]

D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms, Comm. Math. Phys., 213 (2000), 181-201.  doi: 10.1007/s002200000238.

[22]

D. DolgopyatM. Viana and J. Yang, Geometric and measure-theoretical structures of maps with mostly contracting center, Comm. Math. Phys., 341 (2016), 991-1014.  doi: 10.1007/s00220-015-2554-y.

[23]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, in The Theory of Chaotic Attractors, Springer, New York, 1985,273–312. doi: 10.1007/978-0-387-21830-4_17.

[24]

J.-P. Eckmann and L.-S. Young, Nonequilibrium energy profiles for a class of 1-D models, Comm. Math. Phys., 262 (2006), 237-267.  doi: 10.1007/s00220-005-1462-y.

[25]

B. Fernandez, Computer-assisted proof of loss of ergodicity by symmetry breaking in expanding coupled maps, Ann. Henri Poincaré, 21 (2020), 649–674. doi: 10.1007/s00023-019-00876-2.

[26]

T. Fischer and H. H. Rugh, Transfer operators for coupled analytic maps, Ergodic Theory Dynam. Systems, 20 (2000), 109-143.  doi: 10.1017/S0143385700000079.

[27]

D. HadjiloucasM. J. Nicol and C. P. Walkden, Regularity of invariant graphs over hyperbolic systems, Ergodic Theory Dynam. Systems, 22 (2002), 469-482.  doi: 10.1017/S0143385702000226.

[28]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2002.

[29]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.

[30]

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

[31]

K. Kaneko, Theory and Applications of Coupled Map Lattices, Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester, 1993.

[32]

J. L. KaplanJ. Mallet-Paret and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory Dynam. Systems, 4 (1984), 261-281.  doi: 10.1017/S0143385700002431.

[33]

G. Keller and M. Künzle, Transfer operators for coupled map lattices, Ergodic Theory Dynam. Systems, 12 (1992), 297-318.  doi: 10.1017/S0143385700006763.

[34]

G. Keller and C. Liverani, Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Comm. Math. Phys., 262 (2006), 33-50.  doi: 10.1007/s00220-005-1474-7.

[35]

G. Keller and C. Liverani, Map lattices coupled by collisions, Comm. Math. Phys., 291 (2009), 591-597.  doi: 10.1007/s00220-009-0835-z.

[36]

J. Koiller and L.-S. Young, Coupled map networks, Nonlinearity, 23 (2010), 1121-1141.  doi: 10.1088/0951-7715/23/5/006.

[37]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, Berlin, 1975,420–422. doi: 10.1007/BFb0013365.

[38]

F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 163–188.

[39]

F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin entropy formula, Ergodic Theory Dynam. Systems, 2 (1982), 203-219.  doi: 10.1017/S0143385700001528.

[40]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509–539. doi: 10.2307/1971328.

[41]

K. Lu, Q. Wang and L.-S. Young, Strange attractors for periodically forced parabolic equations, Mem. Amer. Math. Soc., 224 (2013). doi: 10.1090/S0065-9266-2012-00669-1.

[42]

T. PereiraS. van Strien and M. Tanzi, Heterogeneously coupled maps: Hub dynamics and emergence across connectivity layers, J. Eur. Math. Soc. (JEMS), 22 (2020), 2183-2252.  doi: 10.4171/JEMS/963.

[43]

Y. B. Pesin and Y. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.

[44]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.

[45]

C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54.  doi: 10.1090/S0002-9947-1989-0983869-1.

[46]

E. R. Pujals and M. Sambarino, On the dynamics of dominated splitting, Ann. of Math. (2), 169 (2009), 675–739. doi: 10.4007/annals.2009.169.675.

[47]

C. Robinson, Structural stability of $C^1$ diffeomorphisms, J. Differential Equations, 22 (1976), 28-73.  doi: 10.1016/0022-0396(76)90004-8.

[48]

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

[49]

D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 219 (2001), 481-487.  doi: 10.1007/s002200100420.

[50]

F. Sélley and P. Bálint, Mean-field coupling of identical expanding circle maps, J. Stat. Phys., 164 (2016), 858-889.  doi: 10.1007/s10955-016-1568-y.

[51]

J. G. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.  doi: 10.1070/RM1972v027n04ABEH001383.

[52]

S. H. Strogatz and I. Stewart, Coupled oscillators and biological synchronization, Scientific Amer., 269 (1993), 102-109.  doi: 10.1038/scientificamerican1293-102.

[53]

M. Tsujii, Regular points for ergodic Sinaĭ measures, Trans. Amer. Math. Soc., 328 (1991), 747-766.  doi: 10.2307/2001802.

[54]

J. D. Wang and P. A. Levin, Metabolism, cell growth and the bacterial cell cycle, Nature Reviews Microbiology, 7 (2009), 822-827.  doi: 10.1038/nrmicro2202.

[55]

Q. Wang and L.-S. Young, From invariant curves to strange attractors, Comm. Math. Phys., 225 (2002), 275-304.  doi: 10.1007/s002200100582.

[56]

Q. Wang and L.-S. Young, Toward a theory of rank one attractors, Ann. of Math. (2), 167 (2008), 349–480. doi: 10.4007/annals.2008.167.349.

[57]

L.-S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. Amer. Math. Soc., 287 (1985), 41-48.  doi: 10.1090/S0002-9947-1985-0766205-1.

[58]

L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.  doi: 10.1090/S0002-9947-1990-0975689-7.

[59]

L.-S. Young, Some open sets of nonuniformly hyperbolic cocycles, Ergodic Theory Dynam. Systems, 13 (1993), 409-415.  doi: 10.1017/S0143385700007446.

[60]

L.-S. Young, Ergodic theory of differentiable dynamical systems, in Real and Complex Dynamical Systems, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995,293–336. doi: 10.1007/978-94-015-8439-5_12.

[61]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2), 147 (1998), 585–650. doi: 10.2307/120960.

[62]

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.

[63]

L.-S. Young, Towards a mathematical model of the brain, J. Statist. Phys., (2020), 1–18. doi: 10.1007/s10955-019-02483-1.

show all references

References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Global dominated splittings and the ${C}^1$ Newhouse phenomenon, Proc. Amer. Math. Soc., 134 (2006), 2229-2237.  doi: 10.1090/S0002-9939-06-08445-0.

[2]

V. Baladi, M. Degli Esposti, S. Isola, E. Järvenpää and A. Kupiainen, The spectrum of weakly coupled map lattices, J. Math. Pures Appl. (9), 77 (1998), 539–584. doi: 10.1016/S0021-7824(98)80138-4.

[3]

P. BálintT. GilbertP. NándoriD. Szász and and I. P. Tóth, On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas, J. Stat. Phys., 166 (2017), 903-925.  doi: 10.1007/s10955-016-1598-5.

[4]

J.-B. BardetG. Keller and and R. Zweimüller, Stochastically stable globally coupled maps with bistable thermodynamic limit, Comm. Math. Phys., 292 (2009), 237-270.  doi: 10.1007/s00220-009-0854-9.

[5]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73–169. doi: 10.2307/2944326.

[6]

M. Benedicks and L.-S. Young, Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576.  doi: 10.1007/BF01232446.

[7]

K. Bjerklöv, A note on circle maps driven by strongly expanding endomorphisms on $\Bbb{T}$, Dyn. Syst., 33 (2018), 361-368.  doi: 10.1080/14689367.2017.1386161.

[8]

S. BoccalettiV. LatoraY. MorenoM. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.

[9]

J. Bochi and M. Viana, Lyapunov Exponents: How frequently are dynamical systems hyperbolic?, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004,271–297.

[10]

C. Bonatti, L. J. Díaz and E. R. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2), 158 (2003), 355–418. doi: 10.4007/annals.2003.158.355.

[11]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585.

[12]

C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508.  doi: 10.1016/j.top.2004.10.009.

[13]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/BFb0081279.

[14]

J. Bricmont and A. Kupiainen, High temperature expansions and dynamical systems, Comm. Math. Phys., 178 (1996), 703-732.  doi: 10.1007/BF02108821.

[15]

M. I. Brin and J. B. Pesin, Partially hyperbolic dynamical systems, Uspehi Mat. Nauk, 28 (1973), 169-170. 

[16]

L. BunimovichC. LiveraniA. Pellegrinotti and Y. Suhov, Ergodic systems of $n$ balls in a billiard table, Comm. Math. Phys., 146 (1992), 357-396.  doi: 10.1007/BF02102633.

[17]

L. A. Bunimovich and Y. G. Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1988), 491-516.  doi: 10.1088/0951-7715/1/4/001.

[18]

J.-R. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics, 671, Springer, Berlin, 2005. doi: 10.1007/b103930.

[19]

H. de Jong, Modeling and simulation of genetic regulatory systems: A literature review, J. Comput. Biol., 9 (2002), 67-103.  doi: 10.1089/10665270252833208.

[20]

L. J. DíazE. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.  doi: 10.1007/BF02392945.

[21]

D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms, Comm. Math. Phys., 213 (2000), 181-201.  doi: 10.1007/s002200000238.

[22]

D. DolgopyatM. Viana and J. Yang, Geometric and measure-theoretical structures of maps with mostly contracting center, Comm. Math. Phys., 341 (2016), 991-1014.  doi: 10.1007/s00220-015-2554-y.

[23]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, in The Theory of Chaotic Attractors, Springer, New York, 1985,273–312. doi: 10.1007/978-0-387-21830-4_17.

[24]

J.-P. Eckmann and L.-S. Young, Nonequilibrium energy profiles for a class of 1-D models, Comm. Math. Phys., 262 (2006), 237-267.  doi: 10.1007/s00220-005-1462-y.

[25]

B. Fernandez, Computer-assisted proof of loss of ergodicity by symmetry breaking in expanding coupled maps, Ann. Henri Poincaré, 21 (2020), 649–674. doi: 10.1007/s00023-019-00876-2.

[26]

T. Fischer and H. H. Rugh, Transfer operators for coupled analytic maps, Ergodic Theory Dynam. Systems, 20 (2000), 109-143.  doi: 10.1017/S0143385700000079.

[27]

D. HadjiloucasM. J. Nicol and C. P. Walkden, Regularity of invariant graphs over hyperbolic systems, Ergodic Theory Dynam. Systems, 22 (2002), 469-482.  doi: 10.1017/S0143385702000226.

[28]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2002.

[29]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.

[30]

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

[31]

K. Kaneko, Theory and Applications of Coupled Map Lattices, Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester, 1993.

[32]

J. L. KaplanJ. Mallet-Paret and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory Dynam. Systems, 4 (1984), 261-281.  doi: 10.1017/S0143385700002431.

[33]

G. Keller and M. Künzle, Transfer operators for coupled map lattices, Ergodic Theory Dynam. Systems, 12 (1992), 297-318.  doi: 10.1017/S0143385700006763.

[34]

G. Keller and C. Liverani, Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Comm. Math. Phys., 262 (2006), 33-50.  doi: 10.1007/s00220-005-1474-7.

[35]

G. Keller and C. Liverani, Map lattices coupled by collisions, Comm. Math. Phys., 291 (2009), 591-597.  doi: 10.1007/s00220-009-0835-z.

[36]

J. Koiller and L.-S. Young, Coupled map networks, Nonlinearity, 23 (2010), 1121-1141.  doi: 10.1088/0951-7715/23/5/006.

[37]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, Berlin, 1975,420–422. doi: 10.1007/BFb0013365.

[38]

F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 163–188.

[39]

F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin entropy formula, Ergodic Theory Dynam. Systems, 2 (1982), 203-219.  doi: 10.1017/S0143385700001528.

[40]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509–539. doi: 10.2307/1971328.

[41]

K. Lu, Q. Wang and L.-S. Young, Strange attractors for periodically forced parabolic equations, Mem. Amer. Math. Soc., 224 (2013). doi: 10.1090/S0065-9266-2012-00669-1.

[42]

T. PereiraS. van Strien and M. Tanzi, Heterogeneously coupled maps: Hub dynamics and emergence across connectivity layers, J. Eur. Math. Soc. (JEMS), 22 (2020), 2183-2252.  doi: 10.4171/JEMS/963.

[43]

Y. B. Pesin and Y. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.

[44]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.

[45]

C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54.  doi: 10.1090/S0002-9947-1989-0983869-1.

[46]

E. R. Pujals and M. Sambarino, On the dynamics of dominated splitting, Ann. of Math. (2), 169 (2009), 675–739. doi: 10.4007/annals.2009.169.675.

[47]

C. Robinson, Structural stability of $C^1$ diffeomorphisms, J. Differential Equations, 22 (1976), 28-73.  doi: 10.1016/0022-0396(76)90004-8.

[48]

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

[49]

D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 219 (2001), 481-487.  doi: 10.1007/s002200100420.

[50]

F. Sélley and P. Bálint, Mean-field coupling of identical expanding circle maps, J. Stat. Phys., 164 (2016), 858-889.  doi: 10.1007/s10955-016-1568-y.

[51]

J. G. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.  doi: 10.1070/RM1972v027n04ABEH001383.

[52]

S. H. Strogatz and I. Stewart, Coupled oscillators and biological synchronization, Scientific Amer., 269 (1993), 102-109.  doi: 10.1038/scientificamerican1293-102.

[53]

M. Tsujii, Regular points for ergodic Sinaĭ measures, Trans. Amer. Math. Soc., 328 (1991), 747-766.  doi: 10.2307/2001802.

[54]

J. D. Wang and P. A. Levin, Metabolism, cell growth and the bacterial cell cycle, Nature Reviews Microbiology, 7 (2009), 822-827.  doi: 10.1038/nrmicro2202.

[55]

Q. Wang and L.-S. Young, From invariant curves to strange attractors, Comm. Math. Phys., 225 (2002), 275-304.  doi: 10.1007/s002200100582.

[56]

Q. Wang and L.-S. Young, Toward a theory of rank one attractors, Ann. of Math. (2), 167 (2008), 349–480. doi: 10.4007/annals.2008.167.349.

[57]

L.-S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. Amer. Math. Soc., 287 (1985), 41-48.  doi: 10.1090/S0002-9947-1985-0766205-1.

[58]

L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.  doi: 10.1090/S0002-9947-1990-0975689-7.

[59]

L.-S. Young, Some open sets of nonuniformly hyperbolic cocycles, Ergodic Theory Dynam. Systems, 13 (1993), 409-415.  doi: 10.1017/S0143385700007446.

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Figure 1.  (Courtesy of Bastien Fernandez) Illustration of a coupled system having an attractor with wild geometry. The {plotted} points represent the intersection of the attractor $ \Lambda $ with a $ W^{cs}- $manifold through one of the points in the attractor. The horizontal and vertical axes indicate approximate directions of $ E^s $ and $ E^c $
Figure 2.  Top: Evolution of a curve in $ W^{cu} $. Here we show the action of the map on a curve $ \gamma_0 $, $ \gamma_0(J) \subset \{z = \mbox{constant}\} $, such that $ \pi_{xy}\gamma_0 $ is contained in a $ W^u_A- $curve. The pictures on the top show the effect of $ \tilde f_1 $, $ \tilde f_2 $, and $ \tilde f_3 $ on $ \gamma_0 $, where $ \tilde f_1 $, $ \tilde f_2 $, and $ \tilde f_3 $ are the lifts of $ f_1 $, $ f_2 $, and $ f_3 $ respectively. Here we assume that $ r(x) = x $. Bottom: action of $ \tilde F^6 $ on a piece of curve $ \gamma_0 $, where $ \tilde F $ is the lift of the map $ F $. These plots highlight the effect of the monotonicity of the coupling
Figure 3.  The pictures above illustrate assumptions (B2) and (B3). On the left is the graph of an example of $ r $: $ r = 0 $ outside of $ I_\epsilon $ and is very steep when $ r(x) $ is in between $ d $ and $ 1-d $. On the right is the graph of an example of $ g $. The highlighted intervals are $ g(I_+)\backslash I_+ $; each component has diameter less than $ d $
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