# American Institue of Mathematical Sciences

2003, 9(2): 443-450. doi: 10.3934/dcds.2003.9.443

## Holomorphic maps for which the unstable manifolds depend on prehistories

 1 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States 2 Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430, United States

Received  September 2001 Revised  April 2002 Published  December 2002

For points $x$ belonging to a basic set $\Lambda$ of an Axiom A holomorphic endomorphism of $\mathbb P^2$, one can construct the local stable manifold $W_{\varepsilon_0}^s(x)$ and the local unstable manifolds $W_{\varepsilon_0}^u(\hat x)$. A priori, $W_{\varepsilon_0}^u(\hat x)$ should depend on the entire prehistory $\hat x$ of $x$. However, all known examples have all their local unstable manifolds depending only on the base point $x$. Therefore a natural problem is to give actual examples where, for different prehistories of points in the basic sets of holomorphic Axiom A maps, we obtain different unstable manifolds. We solve this problem by considering the map $(z^4+\varepsilon w^2, w^4)$ and then also show that, by perturbing $(z^2+c, w^2)$, one can get also maps $f_\varepsilon$ which are injective on $\Lambda_\varepsilon$, their corresponding basic sets, hence the cardinality of the set $(f_\varepsilon|_{\Lambda_\varepsilon})^{-1}(x), x \in \Lambda_\varepsilon$, is not stable under perturbation.
Citation: Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443
 [1] Eugen Mihailescu. Unstable manifolds and Hölder structures associated with noninvertible maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 419-446. doi: 10.3934/dcds.2006.14.419 [2] Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537 [3] John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047 [4] Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1 [5] Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 [6] Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451 [7] C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1) : 1-50. doi: 10.3934/jcd.2017002 [8] Wenxiong Chen, Congming Li. Harmonic maps on complete manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 799-804. doi: 10.3934/dcds.1999.5.799 [9] Ketty A. De Rezende, Mariana G. Villapouca. Discrete Conley Index Theory for Zero Dimensional Basic Sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056 [10] Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617 [11] Evelyn Sander. Hyperbolic sets for noninvertible maps and relations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 339-357. doi: 10.3934/dcds.1999.5.339 [12] Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301 [13] Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663 [14] Xu Zhang, Yuming Shi, Guanrong Chen. Coupled-expanding maps under small perturbations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1291-1307. doi: 10.3934/dcds.2011.29.1291 [15] Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553 [16] Victor Ayala, Adriano Da Silva, Luiz A. B. San Martin. Control systems on flag manifolds and their chain control sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2301-2313. doi: 10.3934/dcds.2017101 [17] 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 [18] Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343 [19] Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175 [20] Bruce Kitchens, Michał Misiurewicz. Omega-limit sets for spiral maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 787-798. doi: 10.3934/dcds.2010.27.787

2016 Impact Factor: 1.099