Sinai Ruelle Bowen Measures Research Articles

Conditional mixing in deterministic chaos

AbstractWhile on the one hand, chaotic dynamical systems can be predicted for all time given exact knowledge of an initial state, they are also in many cases rapidly mixing, meaning that smooth probabilistic information (quantified by measures) on the system’s state has negligible value for predicting the long-term future. However, an understanding of the long-term predictive value of intermediate kinds of probabilistic information is necessary in various physical problems, and largely remains lacking. Of particular interest in data assimilation and linear response theory are the conditional measures of the Sinai–Ruelle–Bowen (SRB) measure on zero sets of general smooth functions of the phase space. In this paper we give rigorous and numerical evidence that such measures generically converge back under the dynamics to the full SRB measures, exponentially quickly. We call this property conditional mixing. While conditional mixing typically cannot be proven from standard transfer operator theory, we will prove that conditional mixing holds in a class of generalized baker’s maps, and demonstrate it numerically in some non-Markovian piecewise hyperbolic maps. Conditional mixing provides a natural limit on the effectiveness of long-term forecasting of chaotic systems via partial observations, and appears key to proving the existence of linear response outside the setting of smooth uniform hyperbolicity.

Invariant family of leaf measures and the Ledrappier–Young property for hyperbolic equilibrium states

AbstractThe manifold M is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$ , and f is a $C^{1+\beta }$ ( $\kern0.3pt\beta>0$ ) diffeomorphism of M. $\varphi $ is a Hölder continuous potential on M. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $\varphi $ ), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class $H_\chi (p)$ , we prove that $\varphi $ admits a local equilibrium state on $H_\chi (p)$ if and only if $\varphi $ is ‘recurrent on $H_\chi (p)$ ’ (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitute as its conditional measures. An immediate corollary is the local product structure of hyperbolic equilibrium states. Second, we prove a Ledrappier–Young property for hyperbolic equilibrium states: if $\varphi $ admits a conformal family of leaf measures and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to $\varphi $ ) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic Sinai–Ruelle–Bowen measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure zero) has conditional measures on local unstable leaves which are absolutely continuous with respect to the Riemannian volume of these leaves.

Entropy formula for systems with inducing schemes

We obtain entropy formulas for Sinai–Ruelle–Bowen (SRB) measures with finite entropy given by inducing schemes. In the first part of the work, we obtain Pesin entropy formula for the class of noninvertible systems whose SRB measures are given by Gibbs-Markov induced maps. In the second part, we obtain Pesin entropy formula for invertible maps whose SRB measures are given by Young sets, taking into account a classical compression technique along the stable direction that allows a reduction of the return map associated with a Young set to a Gibbs-Markov map. In both cases, we give applications of our main results to several classes of dynamical systems with singular sets, where the classical results by Ruelle and Pesin cannot be applied. We also present examples of systems with SRB measures given by inducing schemes for which Ruelle inequality does not hold.

Approximating the linear response of physical chaos

Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle’s formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai–Ruelle–Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto–Sivashinsky.

On the Abundance of Sinai–Ruelle–Bowen Measures

On the Abundance of Sinai–Ruelle–Bowen Measures

SRB measures of singular hyperbolic attractors

<p style='text-indent:20px;'>It is known that hyperbolic maps admitting singularities have at most countably many ergodic Sinai-Ruelle-Bowen (SRB) measures. These maps include the Belykh attractor, the geometric Lorenz attractor, and more general Lorenz-type systems. In this paper, we establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary in general.</p>

Rank-one strange attractors <i>versus</i> heteroclinic tangles

<p style='text-indent:20px;'>We present two different mechanisms for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. The unfolding of each parameter is associated to the emergence of different dynamical scenarios (rank-one attractors and heteroclinic tangles). These relative positions may be determined using the Melnikov method.</p><p style='text-indent:20px;'>Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "lar-ge" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and non-uniformly hyperbolic attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a <i>prevalent</i> phenomenon.</p>

SRB Measures and Young Towers for Surface Diffeomorphisms

We give geometric conditions that are necessary and sufficient for the existence of Sinai–Ruelle–Bowen (SRB) measures for \(C^{1+\alpha }\) surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original method for constructing first return Young towers, proving that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies on a new general result on hyperbolic branches and shadowing for pseudo-orbits in non-uniformly hyperbolic sets which is of independent interest.

SRB measures for pointwise hyperbolic systems on open regions

A diffeomorphism f : M → M is pointwise partially hyperbolic on an open invariant subset N ⊂ M if there is an invariant decomposition TNM = Eu ⊕ Ec ⊕ Es such that Dxf is strictly expanding on E and contracting on E at each x ∈ N. We show that under certain conditions f has unstable and stable manifolds, and admits a finite or an infinite u-Gibbs measure μ. If f is pointwise hyperbolic on N, then μ is a Sinai-Ruelle-Bowen (SRB) measure or an infinite SRB measure. As applications, we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok’s map have the properties.

Continuity of the SRB entropy of convex projective structures

AbstractThe space of convex projective structures has been well studied with respect to the topological entropy. But, to better understand the geometry of the structure, we study the entropy of the Sinai–Ruelle–Bowen measure and show that it is a continuous function on the space of strictly convex real projective structures.

Some statistical properties of almost Anosov diffeomorphisms

Abstract For a kind of almost Anosov diffeomorphisms, we study the relationship among the existence of Sinai-Ruelle-Bowen (SRB) measures, the local differentiability near the indifferent fixed points, and space dimension, where the almost Anosov diffeomorphisms are hyperbolic everywhere except for the indifferent fixed points. As a consequence, there are C2 almost Anosov diffeomorphisms that admit σ-finite (infinite) SRB measures in spaces with dimensions bigger than one; there exist C2 almost Anosov diffeomorphisms with finite SRB measures in spaces with dimensions bigger than three. Further, we obtain the lower and upper polynomial bounds for the decay rates of the correlation functions of the Holder observables for the maps admitting finite SRB measures.

Applications of variable discounting dynamic programming to iterated function systems and related problems

We study existence and uniqueness of the fixed point solutions of a large class of nonlinear variable discounted transfer operators associated to a sequential decision-making process. We establish regularity properties of these solutions, with respect to the immediate return and the variable discount. In addition, we apply our methods to reformulating and solving, in the setting of dynamic programming, some central variational problems on the theory of iterated function systems, Markov decision processes, discrete Aubry–Mather theory, Sinai–Ruelle–Bowen measures, fat solenoidal attractors, and ergodic optimization.

SRB measures in chaotic dynamical systems

In this paper, we present the construction of natural invariant measures so called SRB(Sinai-Ruelle-Bowen) measures by the properties of geometric t-potential and Bowen's equation for the hyperbolic attractors.

Sinai–Ruelle–Bowen measure for normal form map of grazing bifurcations of impact oscillators

Grazing bifurcations are typical non-smooth bifurcations that occur in impact oscillators. They are described by a discrete map with square-root singularities. In this paper we study the structure of a strange attractor which is the closure of the unstable manifold at hyperbolic fixed point of the normal form map for grazing bifurcations of one-degree-of-freedom impact oscillators from the ergodic theoretic point of view. We prove that for some set of values of the parameters this map has an Sinai–Ruelle–Bowen measure which is supported on the strange attractor and is ergodic.

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

We study polygonal billiards with reflection laws contracting the angle of reflection towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic Sinai–Ruelle–Bowen measures whose basins cover a set of full Lebesgue measure.

The Geometric Approach for Constructing Sinai–Ruelle–Bowen Measures

An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric `push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems.

Linear response and periodic points

Given an expanding map of the interval we can associate an absolutely continuous measure. Given an Anosov transformation on a two torus we can associate a Sinai–Ruelle–Bowen measure. In this note we consider first and second derivatives of the change in the average of a reference function. We present an explicit convergent series for these derivatives. In particular, this gives a relatively simple method of computation.

Erratum to: Spectral Triple and Sinai–Ruelle–Bowen Measures

Erratum to: Spectral Triple and Sinai–Ruelle–Bowen Measures

Spatial structure of Sinai–Ruelle–Bowen measures

Sinai–Ruelle–Bowen measures are the only physically observable invariant measures for billiard dynamical systems under small perturbations. These measures are singular, but as it was noted in Bonetto et al. (2012), marginal distributions of spatial and angular coordinates are absolutely continuous. We generalize these facts and provide full mathematical proofs.

Convergence of rare event point processes to the Poisson process for planar billiards

We show that for planar dispersing billiards the distribution of return times is, in the limit, Poisson for metric balls almost everywhere w.r.t. the SRB (Sinai–Ruelle–Bowen) measure. Since the Poincaré return map is piecewise smooth but becomes singular at the boundaries of the partition elements, recent results on the limiting distribution of return times cannot be applied, as they require the maps to have bounded second derivatives everywhere. We first prove the Poisson limiting distribution assuming exponentially decaying correlations. For the case where the correlations decay polynomially, we induce on a subset on which the induced map has exponentially decaying correlations. We then prove a general theorem according to which the limiting return times statistics of the original map and the induced map are the same.