# American Institute of Mathematical Sciences

September  2014, 34(9): 3761-3772. doi: 10.3934/dcds.2014.34.3761

## Shadowing near nonhyperbolic fixed points

 1 Faculty of Mathematics and Mechanics St. Petersburg State University, University av., 28, 198504, St. Petersburg, Russian Federation 2 Faculty of Mathematics and Mechanics, St. Petersburg State University, University av. 28, 198504, St. Petersburg, Russian Federation

Received  February 2013 Revised  December 2013 Published  March 2014

We use Lyapunov type functions to find conditions of finite shadowing in a neighborhood of a nonhyperbolic fixed point of a one-dimensional or two-dimensional homeomorphism or diffeomorphism. A new concept of shadowing in which we control the size of one-step errors is introduced in the case of a nonisolated fixed point.
Citation: Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761
##### References:

show all references

##### References:
 [1] Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355 [2] Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [3] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [4] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [5] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [6] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [7] Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381 [8] Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271 [9] Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115 [10] Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621 [11] Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645 [12] Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467 [13] Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248 [14] Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273 [15] Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial & Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487 [16] Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843 [17] Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793 [18] Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881 [19] Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187 [20] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

2018 Impact Factor: 1.143