Sinai Ruelle Bowen Measures Research Articles

Spectral Triple and Sinai–Ruelle–Bowen Measures

In this article, we recover the Sinai–Ruelle–Bowen measure associated to a real-valued Holder continuous function defined on the Julia set of a hyperbolic quadratic polynomial, as a noncommutative measure by constructing an appropriate spectral triple.

Stability index for chaotically driven concave maps

We study skew product systems driven by a hyperbolic base map S ^ : Θ → Θ (for example, a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on R + like x ↦ g ^ ( θ ) arctan ( x ) , where θ ∈ Θ is a parameter driven by the base map. The fibrewise attractor is the graph of an upper semicontinuous function θ ↦ φ ^ ∞ ( θ ) ∈ R + . For many choices of g ^ , φ ^ ∞ has a residual set of zeros, but φ ^ ∞ > 0 μ SRB -almost surely, where μ SRB is the Sinai–Ruelle–Bowen measure of S ^ - 1 . In such situations, we evaluate the stability index of the global attractor of the system, which is the subgraph { ( θ , x ) ∈ Θ × R + : 0 ⩽ x ⩽ φ ^ ∞ ( θ ) } of φ ^ ∞ , at all regular points ( θ , 0 ) in terms of the local exponents Γ ^ ( θ ) : = lim n → ∞ ( 1 / n ) log g ^ n ( θ ) and Λ ^ ( θ ) : = lim n → ∞ ( 1 / n ) log | D u S ^ - n ( θ ) | and of the positive zero s * of a certain thermodynamic pressure function associated with S ^ and g ^ . (In queuing theory, an analogue of s * is known as Loynes’ exponent [R. M. Loynes, ‘The stability of a queue with non-independent inter-arrival and service times’, Proc. Cambridge Philos. Soc. 58 (1962) 497–520].) The stability index was introduced by Podvigina and Ashwin [‘On local attraction properties and a stability index for heteroclinic connections’, Nonlinearity 24 (2011) 887–929.] to quantify the local scaling of basins of attraction.

Statistical properties of generalized Viana maps

We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana. In particular, we construct a class of multi-dimensional non-uniformly expanding attractors that exhibit both critical points and discontinuities and prove existence and uniqueness of an SRB (Sinai–Ruelle–Bowen) measure with stretched-exponential decay of correlations, stretched-exponential large deviations and satisfying some limit laws. Moreover, generically such maps admit the coexistence of a dense subset of points with negative central Lyapunov exponent together with a full Lebesgue measure subset of points which have positive Lyapunov exponents in all directions. Finally, we discuss the existence of SRB measures for skew-products associated with hyperbolic parameters by the study of fibred hyperbolic maps.

Equilibrium states and SRB-like measures of $C^1$-expanding maps of the circle

For any C^1 -expanding map f of the circle we study the equilibrium states for the potential \psi=-\log |f'| . We formulate a C^1 generalization of Pesin’s Entropy Formula that holds for all the Sinai–Ruelle–Bowen (SRB) measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C^1 -generic case Pesin’s Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non-generic case for which no SRB measure exists.

Strong stochastic stability for non-uniformly expanding maps

AbstractWe consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araújo [Random perturbations of non-uniformly expanding maps.Astérisque 286(2003), 25–62], where the stochastic stability in the$\mathrm {weak}^*$topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai–Ruelle–Bowen (SRB) measure of the unperturbed system in the$L^1$-norm. As an application of our results, we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in Alves, Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly expanding.Invent. Math. 140(2000), 351–398] and the second one is the class ofViana maps.

Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems

AbstractWe study the number of visits to balls Br(x), up to time t/μ(Br(x)), for a class of non-uniformly hyperbolic dynamical systems, where μ is the Sinai–Ruelle–Bowen measure. Outside a set of ‘bad’ centers x, we prove that this number is approximately Poissonnian with a controlled error term. In particular, when r→0, we get convergence to the Poisson law for a set of centers of μ-measure one. Our theorem applies for instance to the Hénon attractor and, more generally, to systems modelled by a Young tower whose return-time function has an exponential tail and with one-dimensional unstable manifolds. Along the way, we prove an abstract Poisson approximation result of independent interest.

Singularities of the susceptibility of a Sinai–Ruelle–Bowen measure in the presence of stable–unstable tangencies

Let ρ be a Sinai-Ruelle-Bowen (SRB or 'physical') measure for the discrete time evolution given by a map f, and let ρ(A) denote the expectation value of a smooth function A. If f depends on a parameter, the derivative δρ(A) of ρ(A) with respect to the parameter is formally given by the value of the so-called susceptibility function Ψ(z) at z=1. When f is a uniformly hyperbolic diffeomorphism, it has been proved that the power series Ψ(z) has a radius of convergence r(Ψ)>1, and that δρ(A)=Ψ(1), but it is known that r(Ψ)<1 in some other cases. One reason why f may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for (f,ρ). The present paper gives a crude, non-rigorous, analysis of this situation in terms of the Hausdorff dimension d of ρ in the stable direction. We find that the tangencies produce singularities of Ψ(z) for |z|<1 if d<1/2, but only for |z|>1 if d>1/2. In particular, if d>1/2, we may hope that Ψ(1) makes sense, and the derivative δρ(A)=Ψ(1) thus has a chance to be defined.

SRB measures for certain Markov processes

We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.

Asymptotic distributions of preimages for endomorphisms

Attractors for hyperbolic diffeomorphisms are known to possess unique Sinai–Ruelle–Bowen measures with interesting properties. In this paper we investigate the case of non-invertible maps (endomorphisms) which have repellers Λ. We work with preimages of points in a neighbourhood of the repeller (assumed to be non-expanding); the situation here is different than the one for diffeomorphisms or positive iterates of endomorphisms. We give two methods to obtain invariant measures from local inverse iterates. We show that if Λ is a hyperbolic s-conformal repeller for f, not necessarily expanding, and if f is d-to-1 on Λ then for Lebesgue almost every x in the repelling basin of Λ there are histories of x asymptotically distributed like the equilibrium measure μs of the Hölder continuous potential Φs, with Φs (y):=log ∣Dfs (y)∣ for y∈Λ. The measure μs plays the role of an inverse Sinai–Ruelle–Bowen measure on the non-invertible repeller. We prove also that there exists a set A⊂Wuε(Λ) with λ(A)=λ(Wuε(Λ)) (where λ(⋅) is the Lebesgue measure) such that for any z∈A and any real continuous function g, with In particular, we obtain the asymptotic distribution of preimages of Lebesgue almost all points for a class of hyperbolic toral endomorphisms on 𝕋m,m≥2 .

Dynamics of homoclinic tangles in periodically perturbed second-order equations

We obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second-order ordinary differential equations with dissipation. Let μ be the size of perturbation and Λ μ be the entire homoclinic tangle. We prove in particular that (i) for infinitely many disjoint open sets of μ, Λ μ contains nothing else but a horseshoe of infinitely many branches; (ii) for infinitely many disjoint open sets of μ, Λ μ contains nothing else but one sink and one horseshoe of infinitely many branches; and (iii) there are positive measure sets of μ so that Λ μ admits strange attractors with Sinai–Ruelle–Bowen measure. We also use the equation d 2 q d t 2 + ( λ − γ q 2 ) d q d t − q + q 2 = μ q 2 sin ω t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation.

Periodic occurrence of chaotic behavior of homoclinic tangles

In this article, we illustrate, through numerical simulations, some important aspects of the dynamics of the periodically perturbed homoclinic solutions for a dissipative saddle. More explicitly, we demonstrate that, when homoclinic tangles are created, three different dynamical phenomena, namely, horseshoes, periodic sinks, and attractors with Sinai–Ruelle–Bowen measures, manifest themselves periodically with respect to the magnitude of the forcing function. In addition, when the stable and the unstable manifolds are pulled apart so as not to intersect, first, rank 1 attractors, then quasi-periodic attractors are added to the dynamical scene.

Alternative proofs of linear response for piecewise expanding unimodal maps

AbstractWe give two new proofs that the Sinai–Ruelle–Bowen (SRB) measure t↦μt of a C2 path ft of unimodal piecewise expanding C3 maps is differentiable at 0 if ft is tangent to the topological class of f0. The arguments are more conceptual than the original proof of Baladi and Smania [Linear response formula for piecewise expanding unimodal maps. Nonlinearity21 (2008), 677–711], but require proving Hölder continuity of the infinitesimal conjugacy α (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order of ∫ ψ dμt if ft is smooth enough and stays in the topological class of f0 and if ψ is smooth enough (a new result). In addition, this proof does not require any information on the decomposition of the SRB measure into regular and singular terms, making it potentially amenable to extensions to higher dimensions. The second new proof allows us to recover the linear response formula (i.e. the formula for the derivative at 0) obtained by Baladi and Smania, by an argument more conceptual than the ‘brute force’ cancellation mechanism used by Baladi and Smania.

Sinai Billiards under Small External Forces II

Consider a particle moving freely on the torus and colliding elastically with some fixed convex bodies. This model is called a periodic Lorentz gas, or a Sinai billiard. It is a Hamiltonian system with a smooth invariant measure, whose ergodic and statistical properties have been well investigated. Now let the particle be subjected to a small external force. This new system is not likely to have a smooth invariant measure. Then a Sinai-Ruelle-Bowen (SRB) measure describes the evolution of typical phase trajectories. We find general sufficient conditions on the external force under which the SRB measure for the collision map exists, is unique, and enjoys good ergodic and statistical properties, including Bernoulliness and an exponential decay of correlations.

MARKOV CHAIN PERTURBATIONS OF A CLASS OF PARTIALLY EXPANDING ATTRACTORS

In this paper Markov chain perturbations of a class of partially expanding attractors of a diffeomorphism are considered. We show that, under some regularity conditions on the transition probabilities, the zero-noise limits of stationary measures of the Markov chains are Sinai–Ruelle–Bowen measures of the diffeomorphism on the attractors.

Random perturbations and statistical properties of Hénon-like maps

For a large class of non-uniformly hyperbolic diffeomorphisms, we prove stochastic stability under small random noise: the unique stationary probability measure of the Markov chain converges to the Sinai–Ruelle–Bowen measure of the unperturbed attractor when the noise level tends to zero.

Sinai–Ruelle–Bowen measures for weakly expanding maps

We construct Sinai–Ruelle–Bowen measures for endomorphisms satisfying conditions far weaker than the usual non-uniform expansion. As a consequence, the definition of a non-uniformly expanding map can be weakened. We also prove the existence of an absolutely continuous invariant measure for local diffeomorphisms, only assuming the existence of hyperbolic times for Lebesgue at almost every point in the manifold.

Basin problem for Hénon-like attractors in arbitrary dimensions

We prove that Hénon-like strange attractors of diffeomorphisms in any dimensions, such as considered in [2], [7], and [9] support a unique Sinai-Ruelle-Bowen (SRB) measure and have the no-hole property: Lebesgue <i>almost every point</i> in the basin of attraction is generic for the SRB measure. This extends two-dimensional results of Benedicks-Young [4] and Benedicks-Viana [3], respectively.

SRB measures as zero-noise limits

We consider zero-noise limits of random perturbations of dynamical systems and examine, in terms of the continuity of entropy and Lyapunov exponents, circumstances under which these limits are Sinai–Ruelle–Bowen (SRB) measures. The ideas of this general discussion are then applied to specific classes of attractors. We prove, for example, that partially hyperbolic attractors with one-dimensional center subbundles always admit SRB measures.

Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps

We consider the quadratic family of maps given by fa(x) = 1 − ax2 on I = [−1, 1], for the Benedicks–Carleson parameters. On this positive Lebesgue measure set of parameters close to a = 2, fa presents an exponential growth of the derivative along the orbit of the critical point and has an absolutely continuous Sinai–Ruelle–Bowen (SRB) invariant measure. We show that the volume of the set of points of I which, at a given time, fail to present an exponential growth of the derivative, decays exponentially as time passes. We also show that the set of points of I that are not slowly recurrent to the critical set decays sub-exponentially. As a consequence, we obtain continuous variation of the SRB measures and associated metric entropies with the parameter on the referred set. For this purpose, we elaborate on the Benedicks–Carleson techniques in the phase space setting.

Stably ergodic diffeomorphisms which are not partially hyperbolic

We show stable ergodicity of a class of conservative diffeomorphisms ofT n which do not have any hyperbolic invariant subbundle. Moreover, the uniqueness of SRB (Sinai-Ruelle-Bowen) measure for non-conservativeC 1 perturbations of such diffeomorphisms is verified. This class strictly contains non-partially hyperbolic robustly transitive diffeomorphisms constructed by Bonatti-Viana [4] and so we answer the question posed there on the stable ergodicity of such systems.