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Recently, a new version of expansiveness which is closely attached to some certain weak version of hyperbolicity was given for $C^1$ vector fields as following: a $C^1$ vector field $X$ will be called {\it rescaling expansive} on a compact invariant set $\Lambda$ of $X$ if for any $\epsilon>0$ there is $\delta>0$ such that, for any $x,y\in \Lambda$ and any time reparametrization $\theta:\mathbb{R}\to \mathbb{R}$, if $d(\varphi_t(x), \varphi_{\theta(t)}(y))\leq \delta\|X(\varphi_t(x))\|$ for all $t\in \mathbb R$, then $\varphi_{\theta(t)}(y)\in \varphi_{(-\epsilon, \epsilon)}(\varphi_t(x))$ for all $t\in \mathbb R$. In this paper, some equivalent definitions for rescaled expansiveness are given.
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참고문헌 (13)
참고문헌 신청
V. Araujo / 2009 / Singular-hyperbolic attractors are chaotic / Trans. Amer. Math. Soc. 361(5):2431~2485
C. Bonatti / 2016 / Weak hyperbolic structures and robust properties of diffeomorphisms and flows / Proceedings of 7ECM
R. Bowen / 1972 / Expansive one-parameter flows / J. Differential Equations 12:180~193
S. Gan / Morse-Smale systems and horseshoes for three dimensional singular flows / Ann. Ec. Norm. Super.
J. Guckenheimer / 1976 / A strange, strange attractor, In The Hopf bifurcation theorem and its applications / Springer Verlag :368~381
C. Bonatti / 2016 / Weak hyperbolic structures and robust properties of diffeomorphisms and flows / Proceedings of 7ECM
R. Bowen / 1972 / Expansive one-parameter flows / J. Differential Equations 12:180~193
S. Gan / Morse-Smale systems and horseshoes for three dimensional singular flows / Ann. Ec. Norm. Super.
J. Guckenheimer / 1976 / A strange, strange attractor, In The Hopf bifurcation theorem and its applications / Springer Verlag :368~381
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