Statistical Properties of One-Dimensional Maps With Critical Points and Singularities

Díaz-Ordaz, Karla; Holland, Martin; Luzzatto, Stefano

doi:10.1142/s0219493706001852

Cited by 35 publications

(74 citation statements)

References 48 publications

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“…Such attractors may also occur in unfoldings from certain homoclinic and heteroclinic bifurcations; see [228,293]. For an analysis of the dynamics of interval maps with both singularities and critical points that yield approximate one-dimensional models for these attractors, see [110,263,264].…”

Section: Theorem 43 ([333]mentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

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An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.

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Section: Theorem 43 ([333]mentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

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“…† The addition of exponent max − 1 for δ in this formula is a correction to [DHL,Condition (H1)], which affects the proofs only in the sense that some constants will be different. † The addition of exponent max − 1 for δ in this formula is a correction to [DHL,Condition (H1)], which affects the proofs only in the sense that some constants will be different.…”

Section: 42mentioning

confidence: 99%

“…(2.5) † The fact that α * c depends on c in this way is a correction to [DHL,Condition (H2)], where this is not stated, but used in the [DHL, proof of Lemma 2].…”

Section: 42mentioning

confidence: 99%

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Existence and convergence properties of physical measures for certain dynamical systems with holes

Bruin

Demers

Melbourne

2009

Ergod. Th. Dynam. Sys.

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We study two classes of dynamical systems with holes: expanding maps of the interval and Collet-Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure $\mu$ (a.c.c.i.m.) with the physical property that strictly positive H\"{o}lder continuous functions converge to the density of $\mu$ under the renormalized dynamics of the system. In addition, we construct an invariant measure $\nu$, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for H\"{o}lder observables. We show that $\nu$ satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet-Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the \accim and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes

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“…Maps with unbounded derivative are of interest due to their links with the Lorenz map. See [32] and [23] for a discussion of this and [1] and [9] for existence results for absolutely continuous, invariant, probability measures. An early, general and very powerful result is by Rychlik [30].…”

Section: Maps With Unbounded Derivativementioning

confidence: 99%

On cusps and flat tops

Dobbs¹

2014

Ann. inst. Fourier

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Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to C 1+ǫ . The critical points are not required to verify a non-flatness condition, so the results are applicable to C 1+ǫ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

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Statistical Properties of One-Dimensional Maps With Critical Points and Singularities

Cited by 35 publications

References 48 publications

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Existence and convergence properties of physical measures for certain dynamical systems with holes

On cusps and flat tops

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