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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 - A, 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.  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 $\mathbfP^k$], 182 (1999), 143.   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 $\mathbfP^k$], 93 (2001), 145.   Google Scholar

[4]

R. Devaney, "An Introduction to Chaotic Dynamical Systems,", Reprint of the 2nd edition, (2003).   Google Scholar

[5]

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

[6]

J. E. Fornæss and N. Sibony, Critically finite rational maps on $\mathbfP^k$,, in, 137 (1992), 245.   Google Scholar

[7]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions,, Notes partially written by Estela A. Gavosto, 439 (1994), 131.   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.   Google Scholar

[9]

J. E. Fornæss and N. Sibony, Dynamics of $\mathbfP^2$ (Examples),, in, 269 (2001), 47.   Google Scholar

[10]

J. E. Fornæss and B. Weickert, Attractors in $\mathbfP^2$,, in, 37 (1999), 1995.   Google Scholar

[11]

P. Griffiths and J. Harris, "Principles of Algebraic Geometry,", Wiley Classics Library, (1994).   Google Scholar

[12]

M. Jonsson, Some properties of 2-critically finite holomorphic maps of $\mathbfP^2$,, Ergodic Theory Dynam. Systems, 18 (1998), 171.  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.  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 (1995).   Google Scholar

[15]

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

[16]

F. Rong, "Critically Finite Maps, Attractors and Local Dynamics,", Ph.D. Thesis, (2007).   Google Scholar

[17]

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

[18]

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

[19]

P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (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.  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 $\mathbfP^k$], 182 (1999), 143.   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 $\mathbfP^k$], 93 (2001), 145.   Google Scholar

[4]

R. Devaney, "An Introduction to Chaotic Dynamical Systems,", Reprint of the 2nd edition, (2003).   Google Scholar

[5]

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

[6]

J. E. Fornæss and N. Sibony, Critically finite rational maps on $\mathbfP^k$,, in, 137 (1992), 245.   Google Scholar

[7]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions,, Notes partially written by Estela A. Gavosto, 439 (1994), 131.   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.   Google Scholar

[9]

J. E. Fornæss and N. Sibony, Dynamics of $\mathbfP^2$ (Examples),, in, 269 (2001), 47.   Google Scholar

[10]

J. E. Fornæss and B. Weickert, Attractors in $\mathbfP^2$,, in, 37 (1999), 1995.   Google Scholar

[11]

P. Griffiths and J. Harris, "Principles of Algebraic Geometry,", Wiley Classics Library, (1994).   Google Scholar

[12]

M. Jonsson, Some properties of 2-critically finite holomorphic maps of $\mathbfP^2$,, Ergodic Theory Dynam. Systems, 18 (1998), 171.  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.  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 (1995).   Google Scholar

[15]

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

[16]

F. Rong, "Critically Finite Maps, Attractors and Local Dynamics,", Ph.D. Thesis, (2007).   Google Scholar

[17]

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

[18]

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

[19]

P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).   Google Scholar

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