Local dimension of ergodic measures for two-dimensional Lorenz transformations

Steinberger, Thomas

doi:10.1017/s0143385700000493

Cited by 7 publications

(10 citation statements)

References 10 publications

(14 reference statements)

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“…When the above limits coincide for µ-almost every point, we say that µ is exact dimensional and set d µ = d µ (x) = d µ (x). From the main results of [15,14,46] it follows that in the kind of systems mentioned above (under mild extra conditions on the second coordinate G of F ) µ F is exact dimensional, and the logarithm law holds.…”

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confidence: 95%

“…Given a flow having a singular hyperbolic attractor, there exists a finite family Ξ of welladapted cross-sections of the flow where we can define a Poincaré return map F which satisfies the properties in the statement of Theorem A after a suitable choice of coordinates. To take advantage of the above cited result from Steinberger [46] on exact dimensionality and obtain Proposition 1, we need that the Poincaré return map F be injective, which is not evident in the construction and choice of these cross-sections done at [7]. Because of this, the construction presented here is slightly different from the ones presented elsewhere.…”

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confidence: 99%

“…In Section 6 we also show that the physical measure of the flow is exact dimensional. This is done using the results of [46].…”

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confidence: 99%

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Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

2013

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We consider maps preserving a foliation which is uniformly contracting and a onedimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables.We use this result to deduce exponential decay of correlations for suitable Poincaré maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows. -hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm law. V.A. and M.J.P. were partially supported by CNPq, PRONEX-Dyn.Syst., FAPERJ, Balzan Research Project of J.Palis .

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Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

2013

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“…We recall a result of Steinberger [2000] about the local dimension of Lorenz-like systems and prove that for the singular-hyperbolic system the local dimension is defined at almost every point.…”

Section: Exact Dimensionalitymentioning

confidence: 99%

“…This construction shows that there exists a finite family Ξ of well-adapted cross-sections of the flow on the attractor where we can define a Poincaré return map F which satisfies the properties in the statement of Theorem 7 after a suitable choice of coordinates. We remark that, to take advantage of a result from [Steinberger, 2000] on exact dimensionality of certain classes of measures, we need that the Poincaré return map F be injective, which is not evident in the construction and choice of these cross-sections at [Araújo et al, 2009]. Because of this, the construction presented here is slightly different; see Sec.…”

Section: Decay Of Correlations For the Poincaré Return Map Of Singulamentioning

confidence: 99%

Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

Araújo

Galatolo

Pacífico

2014

Int. J. Bifurcation Chaos

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We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria.We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure.We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenzlike) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz 1430028-1 Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com by THE UNIVERSITY OF WAIKATO on 11/16/14. For personal use only. V. Araujo et al.attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.

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Hausdorff dimension of attractors for two dimensional Lorenz transformations

Steinberger

2000

Isr. J. Math.

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Local dimension of ergodic measures for two-dimensional Lorenz transformations

Abstract: A class of transformations on $[0,1]^2$, which includes transformations obtained by a Poincaré section of the Lorenz equation, is considered. We prove a formula which connects local dimension, entropy and characteristic exponents of ergodic invariant probability measures.

Cited by 7 publications

References 10 publications

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

Hausdorff dimension of attractors for two dimensional Lorenz transformations

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