Cone-fields without constant orbit core dimension

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

Cone-fields without constant orbit core dimension

Łukasz Struski ^1, , Jacek Tabor ^1, and Tomasz Kułaga ^1,

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

Abstract
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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.

Keywords: Cone-field, dominated splitting, expansive map, shadowing property., hyperbolicity.

Mathematics Subject Classification: Primary: 37D2.

Citation: Łukasz Struski, Jacek Tabor, Tomasz Kułaga. Cone-fields without constant orbit core dimension. Discrete and Continuous Dynamical Systems, 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), 419-431.
[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), 641-670. doi: 10.3934/dcds.2011.30.641.
[3]	S. Hruska, A numerical method for constructing the hyperbolic structure of comple Hénon mappings, Found. Comput. Math., 6 (2006), 427-455.
[4]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
[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}., ().
[6]	J. Lewowicz, Lyapunov functions and topological stability, Journal of Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.
[7]	J. Lewowicz, Persistence of semi-trajectories, Journal of Dynamics and Differential Equation, 18 (2006), 1095-1102. doi: 10.1007/s10884-006-9047-9.
[8]	S. Newhouse, Cone-fields, domination, and hyperbolicity, in "Modern Dynamical Systems and Applications" (eds. B. Hasselblatt, M. Brin and Y. Pesin), Cambridge University Press, New York, 2004, 419-433.

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References:

[1]	P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings, Journal of Nonlinear Science, 5 (1995), 419-431.
[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), 641-670. doi: 10.3934/dcds.2011.30.641.
[3]	S. Hruska, A numerical method for constructing the hyperbolic structure of comple Hénon mappings, Found. Comput. Math., 6 (2006), 427-455.
[4]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
[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}., ().
[6]	J. Lewowicz, Lyapunov functions and topological stability, Journal of Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.
[7]	J. Lewowicz, Persistence of semi-trajectories, Journal of Dynamics and Differential Equation, 18 (2006), 1095-1102. doi: 10.1007/s10884-006-9047-9.
[8]	S. Newhouse, Cone-fields, domination, and hyperbolicity, in "Modern Dynamical Systems and Applications" (eds. B. Hasselblatt, M. Brin and Y. Pesin), Cambridge University Press, New York, 2004, 419-433.

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