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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

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