October  2012, 32(10): 3651-3664. doi: 10.3934/dcds.2012.32.3651

Cone-fields without constant orbit core dimension

1. 

Jagiellonian University, Faculty of Mathematics and Computer Science, Łojasiewicza 6, 30-348 Kraków, Poland, Poland, Poland

Received  May 2011 Revised  August 2012 Published  May 2012

As is well-known, the existence of a cone-field with constant orbit core dimension is, roughly speaking, equivalent to hyperbolicity, and consequently guarantees expansivity and shadowing. In this paper we study the case when the given cone-field does not have the constant orbit core dimension. It occurs that we still obtain expansivity even in general metric spaces.
Main Result. Let $X$ be a metric space and let $f:X \rightharpoonup X$ be a given partial map. If there exists a uniform cone-field on $X$ such that $f$ is cone-hyperbolic, then $f$ is uniformly expansive, i.e. there exists $N \in \mathbb{N}$, $\lambda \in [0,1)$ and $\epsilon > 0$ such that for all orbits $\mathrm{x},\mathrm{v}:{-N,\ldots,N} \to X$ \[ d_{\sup}(\mathrm{x},\mathrm{v}) \leq \epsilon \Longrightarrow d(\mathrm{x}_0,\mathrm{v}_0) \leq \lambda d_{\sup}(\mathrm{x},\mathrm{v}). \] } We also show a simple example of a cone hyperbolic orbit in $\mathbb{R}^3$ which does not have the shadowing property.
Citation: Łukasz Struski, Jacek Tabor, Tomasz Kułaga. Cone-fields without constant orbit core dimension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3651-3664. doi: 10.3934/dcds.2012.32.3651
References:
[1]

P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings,, Journal of Nonlinear Science, 5 (1995), 419. Google Scholar

[2]

M. J. Capiński and P. Zgliczyński, Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds,, Discrete and Continuous Dynamical Systems, 30 (2011), 641. doi: 10.3934/dcds.2011.30.641. Google Scholar

[3]

S. Hruska, A numerical method for constructing the hyperbolic structure of comple Hénon mappings,, Found. Comput. Math., 6 (2006), 427. Google Scholar

[4]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by Katok and Leonardo Mendoza, 54 (1995). Google Scholar

[5]

T. Kułaga and J. Tabor, Hyperbolic dynamics in graph-directed IFS, 2010., Available from: \url{http://www2.im.uj.edu.pl/badania/preprinty/imuj2010/pr1009.pdf}., (). Google Scholar

[6]

J. Lewowicz, Lyapunov functions and topological stability,, Journal of Differential Equations, 38 (1980), 192. doi: 10.1016/0022-0396(80)90004-2. Google Scholar

[7]

J. Lewowicz, Persistence of semi-trajectories,, Journal of Dynamics and Differential Equation, 18 (2006), 1095. doi: 10.1007/s10884-006-9047-9. Google Scholar

[8]

S. Newhouse, Cone-fields, domination, and hyperbolicity,, in, (2004), 419. Google Scholar

show all references

References:
[1]

P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings,, Journal of Nonlinear Science, 5 (1995), 419. Google Scholar

[2]

M. J. Capiński and P. Zgliczyński, Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds,, Discrete and Continuous Dynamical Systems, 30 (2011), 641. doi: 10.3934/dcds.2011.30.641. Google Scholar

[3]

S. Hruska, A numerical method for constructing the hyperbolic structure of comple Hénon mappings,, Found. Comput. Math., 6 (2006), 427. Google Scholar

[4]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by Katok and Leonardo Mendoza, 54 (1995). Google Scholar

[5]

T. Kułaga and J. Tabor, Hyperbolic dynamics in graph-directed IFS, 2010., Available from: \url{http://www2.im.uj.edu.pl/badania/preprinty/imuj2010/pr1009.pdf}., (). Google Scholar

[6]

J. Lewowicz, Lyapunov functions and topological stability,, Journal of Differential Equations, 38 (1980), 192. doi: 10.1016/0022-0396(80)90004-2. Google Scholar

[7]

J. Lewowicz, Persistence of semi-trajectories,, Journal of Dynamics and Differential Equation, 18 (2006), 1095. doi: 10.1007/s10884-006-9047-9. Google Scholar

[8]

S. Newhouse, Cone-fields, domination, and hyperbolicity,, in, (2004), 419. Google Scholar

[1]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006

[2]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901

[3]

Keonhee Lee, Manseob Lee. Hyperbolicity of $C^1$-stably expansive homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1133-1145. doi: 10.3934/dcds.2010.27.1133

[4]

Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555

[5]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421

[6]

Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123

[7]

Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205

[8]

Piotr Kościelniak, Marcin Mazur, Piotr Oprocha, Paweł Pilarczyk. Shadowing is generic---a continuous map case. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3591-3609. doi: 10.3934/dcds.2014.34.3591

[9]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087

[10]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237

[11]

Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963

[12]

Pau Martín, David Sauzin, Tere M. Seara. Exponentially small splitting of separatrices in the perturbed McMillan map. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 301-372. doi: 10.3934/dcds.2011.31.301

[13]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231

[14]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089

[15]

Alberto Maspero, Beat Schaad. One smoothing property of the scattering map of the KdV on $\mathbb{R}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1493-1537. doi: 10.3934/dcds.2016.36.1493

[16]

Yoshikazu Giga, Hirotoshi Kuroda. A counterexample to finite time stopping property for one-harmonic map flow. Communications on Pure & Applied Analysis, 2015, 14 (1) : 121-125. doi: 10.3934/cpaa.2015.14.121

[17]

Li-Xia Liu, Sanyang Liu, Chun-Feng Wang. Smoothing Newton methods for symmetric cone linear complementarity problem with the Cartesian $P$/$P_0$-property. Journal of Industrial & Management Optimization, 2011, 7 (1) : 53-66. doi: 10.3934/jimo.2011.7.53

[18]

Cheng Wang, Jian-Guo Liu. Positivity property of second-order flux-splitting schemes for the compressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 201-228. doi: 10.3934/dcdsb.2003.3.201

[19]

Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589

[20]

Narcisse Batangouna, Morgan Pierre. Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system. Communications on Pure & Applied Analysis, 2018, 17 (1) : 1-19. doi: 10.3934/cpaa.2018001

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]