TY - JOUR
T1 - Hadamard-Perron theorems and effective hyperbolicity
AU - Climenhaga, Vaughn
AU - Pesin, Yakov
N1 - Funding Information:
The authors were partially supported by NSF grant 0754911. Ya.P. is partially supported by NSF grant 1101165. V.C. was supported by an NSERC postdoctoral fellowship.
Publisher Copyright:
© Cambridge University Press, 2014.
PY - 2014/11/17
Y1 - 2014/11/17
N2 - We prove several new versions of the Hadamard-Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard-Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of 'effective hyperbolicity' and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.
AB - We prove several new versions of the Hadamard-Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard-Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of 'effective hyperbolicity' and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.
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U2 - 10.1017/etds.2014.49
DO - 10.1017/etds.2014.49
M3 - Article
AN - SCOPUS:84983208653
SN - 0143-3857
VL - 760
SP - 23
EP - 63
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
ER -