# American Institute of Mathematical Sciences

2003, 2003(Special): 834-841. doi: 10.3934/proc.2003.2003.834

## True laminations for complex Hènon maps

 1 Mathematics Department Long Island University, Broolyn Campus, University Plaza, Brooklyn, NY 11201, United States

Received  August 2002 Revised  April 2003 Published  April 2003

In this paper, we are specially interested in the lamination structure for a polynomial diffeomorphism $f$ of $\mathbb(C)^2$ that are conjugate to a finite decomposition of generalized complex Hènon maps on $\mathbb(C)$. We prove that there are true $f$-invariant contracting and expanding measured Riemann surface laminations–injected into the stable and unstable partitions $W^(s/u)$. Leaves of the laminations are conformally isomorphic to the complex plane $\mathbb(C)$. The new ingredients here are the countable collection of the Pesin boxes and a $\sigma$-finite topology, the ‘entropy topology’ on the transversals, defined by the logarithm of the measures obtained by conditioning the unique ergodic measure of maximal entropy $\mu$.
Citation: Meiyu Su. True laminations for complex Hènon maps. Conference Publications, 2003, 2003 (Special) : 834-841. doi: 10.3934/proc.2003.2003.834
 [1] Eugen Mihailescu. Unstable manifolds and Hölder structures associated with noninvertible maps. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 419-446. doi: 10.3934/dcds.2006.14.419 [2] Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4619-4635. doi: 10.3934/dcds.2016001 [3] Leandro Arosio, Anna Miriam Benini, John Erik Fornæss, Han Peters. Dynamics of transcendental Hénon maps III: Infinite entropy. Journal of Modern Dynamics, 2021, 17: 465-479. doi: 10.3934/jmd.2021016 [4] Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451 [5] Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443 [6] Raoul-Martin Memmesheimer, Marc Timme. Stable and unstable periodic orbits in complex networks of spiking neurons with delays. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1555-1588. doi: 10.3934/dcds.2010.28.1555 [7] Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Corrigendum to "Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology". Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4585-4586. doi: 10.3934/dcds.2017196 [8] M. R. S. Kulenović, Orlando Merino. A global attractivity result for maps with invariant boxes. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 97-110. doi: 10.3934/dcdsb.2006.6.97 [9] Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499 [10] Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421 [11] Robert Brooks and Eran Makover. The first eigenvalue of a Riemann surface. Electronic Research Announcements, 1999, 5: 76-81. [12] Matilde Martínez, Shigenori Matsumoto, Alberto Verjovsky. Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem. Journal of Modern Dynamics, 2016, 10: 113-134. doi: 10.3934/jmd.2016.10.113 [13] Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2139-2154. doi: 10.3934/cpaa.2021061 [14] Marina Gonchenko, Sergey Gonchenko, Klim Safonov. Reversible perturbations of conservative Hénon-like maps. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1875-1895. doi: 10.3934/dcds.2020343 [15] Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007 [16] Khashayar Filom, Kevin M. Pilgrim. On the non-monotonicity of entropy for a class of real quadratic rational maps. Journal of Modern Dynamics, 2020, 16: 225-254. doi: 10.3934/jmd.2020008 [17] Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911 [18] Erica Clay, Boris Hasselblatt, Enrique Pujals. Desingularization of surface maps. Electronic Research Announcements, 2017, 24: 1-9. doi: 10.3934/era.2017.24.001 [19] Wacław Marzantowicz, Feliks Przytycki. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501 [20] Ruediger Landes. Stable and unstable initial configuration in the theory wave fronts. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 797-808. doi: 10.3934/dcdss.2012.5.797

Impact Factor: