A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

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Hadamard–Perron theorems and effective hyperbolicity

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.49 ◽

2014 ◽

Vol 36 (1) ◽

pp. 23-63 ◽

Cited By ~ 3

Author(s):

VAUGHN CLIMENHAGA ◽

YAKOV PESIN

Keyword(s):

Closing Lemma ◽

Stable And Unstable Manifolds ◽

Local Dynamics ◽

Local Diffeomorphisms ◽

Finite Amount ◽

Amount Of Information ◽

Unstable Manifolds

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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Quasi-shadowing for partially hyperbolic diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.126 ◽

2014 ◽

Vol 35 (2) ◽

pp. 412-430 ◽

Cited By ~ 8

Author(s):

HUYI HU ◽

YUNHUA ZHOU ◽

YUJUN ZHU

Keyword(s):

Spectral Decomposition ◽

Hyperbolic Systems ◽

Decomposition Theorem ◽

Closing Lemma ◽

Shadowing Property ◽

Hyperbolic Diffeomorphisms ◽

Uniformly Hyperbolic Systems ◽

Partially Hyperbolic ◽

Partially Hyperbolic Diffeomorphisms ◽

Partially Hyperbolic Diffeomorphism

AbstractA partially hyperbolic diffeomorphism $f$ has the quasi-shadowing property if for any pseudo orbit $\{x_{k}\}_{k\in \mathbb{Z}}$, there is a sequence of points $\{y_{k}\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_{k})$ by a motion ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^{0}$-perturbation. When $f$ has a uniformly compact $C^{1}$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems which hold for uniformly hyperbolic systems, such as the Anosov closing lemma, the cloud lemma and the spectral decomposition theorem.

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The Closing Lemma

American Journal of Mathematics ◽

10.2307/2373413 ◽

1967 ◽

Vol 89 (4) ◽

pp. 956 ◽

Cited By ~ 101

Author(s):

Charles C. Pugh

Keyword(s):

Closing Lemma

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The planar closing lemma for chain recurrence

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1994-1107028-9 ◽

1994 ◽

Vol 341 (1) ◽

pp. 173-192 ◽

Cited By ~ 4

Author(s):

Maria Lúcia Alvarenga Peixoto ◽

Charles Chapman Pugh

Keyword(s):

Closing Lemma ◽

Chain Recurrence

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The C1 closing lemma for non-singular endomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700006210 ◽

1991 ◽

Vol 11 (2) ◽

pp. 393-412 ◽

Cited By ~ 9

Author(s):

Lan Wen

Keyword(s):

Closing Lemma

AbstractA slightly improved version of the (idealized) C1 closing lemma is proved. This turns out to generalize the C1 closing lemma from diffeomorphisms to nonsingular endomorphisms.

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