• Previous Article
    Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain
  • DCDS Home
  • This Issue
  • Next Article
    Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets
March  2012, 32(3): 977-989. doi: 10.3934/dcds.2012.32.977

Non-algebraic attractors on $\mathbf{P}^k$

1. 

Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai, 200240, China

Received  September 2010 Revised  May 2011 Published  October 2011

We show that special perturbations of a particular holomorphic map on $\mathbf{P}^k$ give us examples of maps that possess chaotic non-algebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we construct invariant hyperbolic measures supported on the attractors with nice dynamical properties.
Citation: Feng Rong. Non-algebraic attractors on $\mathbf{P}^k$. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 977-989. doi: 10.3934/dcds.2012.32.977
References:
[1]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334. doi: 10.2307/2324899.

[2]

J.-V. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de $\mathbfP^k$, (French) [Lyapunov exponents and distribution of periodic points of an endomorphism on $\mathbfP^k$], Acta Math., 182 (1999), 143-157.

[3]

J.-V. Briend and J. Duval, Deux caractérisations de la mesure d'équilibre d'un endomorphisme de $\mathbfP^k$, (French) [Two characterizations of the equilibrium measure of an endomorphism on $\mathbfP^k$], Publ. Math. Inst. Hautes Études Sci., 93 (2001), 145-159.

[4]

R. Devaney, "An Introduction to Chaotic Dynamical Systems," Reprint of the 2nd edition, Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.

[5]

T.-C. Dinh, Attracting current and equilibrium measure for attractors on $\mathbfP^k$, J. Geom. Anal., 17 (2007), 227-244.

[6]

J. E. Fornæss and N. Sibony, Critically finite rational maps on $\mathbfP^k$, in "The Madison Symposium on Complex Analysis" (Madison, WI, 1991), Contemp. Math., 137, Amer. Math. Soc., Providence, RI, (1992), 245-260.

[7]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions, Notes partially written by Estela A. Gavosto, in "Complex Potential Theory" (eds. P. M. Gauthier and G. Sabidussi) (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Kluwer Academic Publishers, Dordrecht, (1994), 131-186.

[8]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimension. I. Complex analytic methods in dynamical systems, Astérisque, 222 (1994), 201-231.

[9]

J. E. Fornæss and N. Sibony, Dynamics of $\mathbfP^2$ (Examples), in "Laminations and Foliations in Dynamics, Geometry and Topology" (Stony Brook, NY, 1998), Contemp. Math., 269, Amer. Math. Soc., Providence, RI, (2001), 47-85.

[10]

J. E. Fornæss and B. Weickert, Attractors in $\mathbfP^2$, in "Several Complex Variables" (Berkeley, CA, 1995-1996), Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, (1999), 297-307.

[11]

P. Griffiths and J. Harris, "Principles of Algebraic Geometry," Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994.

[12]

M. Jonsson, Some properties of 2-critically finite holomorphic maps of $\mathbfP^2$, Ergodic Theory Dynam. Systems, 18 (1998), 171-187. doi: 10.1017/S0143385798097521.

[13]

M. Jonsson and B. Weickert, A nonalgebraic attractor in $\mathbfP^2$, Proc. Amer. Math. Soc., 128 (2000), 2999-3002. doi: 10.1090/S0002-9939-00-05529-5.

[14]

A. Katok and B. Hasselblatt, "Introduction to The Modern Theory of Dynamical Systems," Encycl. of Math. and its Appl., 54, Cambridge Univ. Press, Cambridge, 1995.

[15]

J. Milnor, On the concept of attractor, Commun. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280.

[16]

F. Rong, "Critically Finite Maps, Attractors and Local Dynamics," Ph.D. Thesis, University of Michigan, 2007.

[17]

F. Rong, The Fatou set for critically finite maps, Proc. Amer. Math. Soc., 136 (2008), 3621-3625. doi: 10.1090/S0002-9939-08-09358-1.

[18]

D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory," Academic Press, Inc., Boston, 1989.

[19]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

show all references

References:
[1]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334. doi: 10.2307/2324899.

[2]

J.-V. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de $\mathbfP^k$, (French) [Lyapunov exponents and distribution of periodic points of an endomorphism on $\mathbfP^k$], Acta Math., 182 (1999), 143-157.

[3]

J.-V. Briend and J. Duval, Deux caractérisations de la mesure d'équilibre d'un endomorphisme de $\mathbfP^k$, (French) [Two characterizations of the equilibrium measure of an endomorphism on $\mathbfP^k$], Publ. Math. Inst. Hautes Études Sci., 93 (2001), 145-159.

[4]

R. Devaney, "An Introduction to Chaotic Dynamical Systems," Reprint of the 2nd edition, Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.

[5]

T.-C. Dinh, Attracting current and equilibrium measure for attractors on $\mathbfP^k$, J. Geom. Anal., 17 (2007), 227-244.

[6]

J. E. Fornæss and N. Sibony, Critically finite rational maps on $\mathbfP^k$, in "The Madison Symposium on Complex Analysis" (Madison, WI, 1991), Contemp. Math., 137, Amer. Math. Soc., Providence, RI, (1992), 245-260.

[7]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions, Notes partially written by Estela A. Gavosto, in "Complex Potential Theory" (eds. P. M. Gauthier and G. Sabidussi) (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Kluwer Academic Publishers, Dordrecht, (1994), 131-186.

[8]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimension. I. Complex analytic methods in dynamical systems, Astérisque, 222 (1994), 201-231.

[9]

J. E. Fornæss and N. Sibony, Dynamics of $\mathbfP^2$ (Examples), in "Laminations and Foliations in Dynamics, Geometry and Topology" (Stony Brook, NY, 1998), Contemp. Math., 269, Amer. Math. Soc., Providence, RI, (2001), 47-85.

[10]

J. E. Fornæss and B. Weickert, Attractors in $\mathbfP^2$, in "Several Complex Variables" (Berkeley, CA, 1995-1996), Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, (1999), 297-307.

[11]

P. Griffiths and J. Harris, "Principles of Algebraic Geometry," Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994.

[12]

M. Jonsson, Some properties of 2-critically finite holomorphic maps of $\mathbfP^2$, Ergodic Theory Dynam. Systems, 18 (1998), 171-187. doi: 10.1017/S0143385798097521.

[13]

M. Jonsson and B. Weickert, A nonalgebraic attractor in $\mathbfP^2$, Proc. Amer. Math. Soc., 128 (2000), 2999-3002. doi: 10.1090/S0002-9939-00-05529-5.

[14]

A. Katok and B. Hasselblatt, "Introduction to The Modern Theory of Dynamical Systems," Encycl. of Math. and its Appl., 54, Cambridge Univ. Press, Cambridge, 1995.

[15]

J. Milnor, On the concept of attractor, Commun. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280.

[16]

F. Rong, "Critically Finite Maps, Attractors and Local Dynamics," Ph.D. Thesis, University of Michigan, 2007.

[17]

F. Rong, The Fatou set for critically finite maps, Proc. Amer. Math. Soc., 136 (2008), 3621-3625. doi: 10.1090/S0002-9939-08-09358-1.

[18]

D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory," Academic Press, Inc., Boston, 1989.

[19]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[1]

Dominic Veconi. SRB measures of singular hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3415-3430. doi: 10.3934/dcds.2022020

[2]

David Parmenter, Mark Pollicott. Gibbs measures for hyperbolic attractors defined by densities. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022038

[3]

Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341

[4]

Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164

[5]

Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505

[6]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811

[7]

Vítor Araújo, Ali Tahzibi. Physical measures at the boundary of hyperbolic maps. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 849-876. doi: 10.3934/dcds.2008.20.849

[8]

B. San Martín, Kendry J. Vivas. Asymptotically sectional-hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4057-4071. doi: 10.3934/dcds.2019163

[9]

Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451

[10]

Anatole Katok. Hyperbolic measures and commuting maps in low dimension. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 397-411. doi: 10.3934/dcds.1996.2.397

[11]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[12]

Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177

[13]

Vaughn Climenhaga, Yakov Pesin, Agnieszka Zelerowicz. Equilibrium measures for some partially hyperbolic systems. Journal of Modern Dynamics, 2020, 16: 155-205. doi: 10.3934/jmd.2020006

[14]

A. M. López. Finiteness and existence of attractors and repellers on sectional hyperbolic sets. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 337-354. doi: 10.3934/dcds.2017014

[15]

Aubin Arroyo, Enrique R. Pujals. Dynamical properties of singular-hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 67-87. doi: 10.3934/dcds.2007.19.67

[16]

V. V. Chepyzhov, A. Miranville. Trajectory and global attractors of dissipative hyperbolic equations with memory. Communications on Pure and Applied Analysis, 2005, 4 (1) : 115-142. doi: 10.3934/cpaa.2005.4.115

[17]

Luis Barreira, Yakov Pesin and Jorg Schmeling. On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture. Electronic Research Announcements, 1996, 2: 69-72.

[18]

Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5133-5152. doi: 10.3934/dcds.2015.35.5133

[19]

Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313

[20]

Zhicong Liu. SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1545-1562. doi: 10.3934/dcds.2013.33.1545

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]