Unique Physical Measure Research Articles

Physical measures for ash attractors

In this paper we will prove that any asymptotically sectional-hyperbolic attractor associated to a three-dimensional C2 vector field X supports an unique physical measure. Asymptotic sectional-hyperbolicity means that any point outside of the stable manifolds of the singularities has arbitrarily large hyperbolic times. Besides, we will describe its main properties and we will deduce some dynamical consequences of the existence of this measure.

Historic behaviour vs. physical measures for irrational flows with multiple stopping points

We study Birkhoff averages along trajectories of smooth reparameterizations of irrational linear flows of the two torus with two stopping points, say p and q, of quadratic order. The limiting behaviour of such averages is independent of the starting point in a set of full Haar-Lebesgue measure and depends in an intricate way on the Diophantine properties of both the slope α of the linear flow as well as the relative position of p and q. In particular, if α is Diophantine, then Birkhoff limits diverge almost everywhere (historic behaviour) and if α is sufficiently Liouville, then there exists some p and q such that the Birkhoff averages converge almost everywhere (unique physical measure).

Parametrized family of pseudo-arc attractors: Physical measures and prime end rotations.

The main goal of this paper is to study topological and measure‐theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure‐preserving map f generates the pseudo‐arc as inverse limit with f as a single bonding map. These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure‐preserving maps f the background Oxtoby–Ulam measures induced by Lebesgue measure for f on the interval are physical on the disc and in addition there is a dense set of maps f defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure‐preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo‐arc attractors with prime ends rotation numbers varying continuously in [0,1/2]. It follows that there are uncountably many dynamically non‐equivalent embeddings of the pseudo‐arc in this family of attractors.

Random composition of L-S-V maps sampled over large parameter ranges

Liverani–Saussol–Vaienti (L–S–V) maps form a family of piecewise differentiable dynamical systems on [0, 1] depending on one parameter . These maps are everywhere expanding apart from a neutral fixed point. It is well known that depending on the amount of expansion close to the neutral point, they have either an absolutely continuous invariant probability measure and polynomial decay of correlations (ω < 1), or a unique physical measure that is singular and concentrated at the neutral point (ω > 1). In this paper, we study the composition of L–S–V maps whose parameters are randomly sampled from a range in , and where these two contrasting behaviours are mixed. We show that if the parameters ω < 1 are sampled with positive probability, then the stationary measure of the random system is absolutely continuous; the annealed decay rate of correlations is close (or in some cases equal) to the fastest rate of decay among those of the sampled systems; and suitably rescaled Birkhoff averages converge to limit laws. In contrast to previous studies where ω ∈ [0, 1], we allow ω > 1 in our sampling distribution. We also show that one can obtain similar decay of correlation rates for , when sampling is done with respect to a family of smooth, heavy-tailed distributions.

Statistical stability of mostly expanding diffeomorphisms

We study how physical measures vary with the underlying dynamics in the open class of Cr, r>1, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs u-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics.A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs cu-states. Both of these may be of independent interest.The non-transitive case is also treated: here the number of physical measures varies upper semi-continuously with the diffeomorphism, and physical measures vary continuously whenever possible.

A robustly transitive diffeomorphism of Kan's type

We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admit two physical measures with intermingled basins. In particular, all these diffeomorphisms are not topologically mixing. Moreover, every such example exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.

Cherry flow: physical measures and perturbation theory

In this article we consider Cherry flows on the torus which have two singularities, a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by Saghin and Vargas [Invariant measures for Cherry flows.Comm. Math. Phys.317(1) (2013), 55–67]. We also show that the perturbation of Cherry flows depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to one of the following three cases: it has a saddle connection; it is a Cherry flow; it is a Morse–Smale flow whose non-wandering set consists of two singularities and one periodic sink. In contrast, when the divergence is non-negative, this flow can be approximated by a non-hyperbolic flow with an arbitrarily large number of periodic sinks.

Robust Exponential Decay of Correlations for Singular-Flows

We construct open sets of C k (k ≥ 2) vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure. In particular we prove that the geometric Lorenz attractor has exponential decay of correlations with respect to the unique physical measure.

Strange Non-Chaotic Attractors in Quasiperiodically Forced Circle Maps

The occurrence of strange non-chaotic attractors (SNA) in quasiperiodically forced systems has attracted considerable interest over the last two decades, in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos. Their existence was first described by Millions̆c̆ikov (and later by Vinograd and also Herman) for quasiperiodic $${\rm SL(2, {\mathbb R})}$$ -cocycles and by Grebogi et al (and later Keller) for so-called pinched skew products. However, except for these two particular classes there are still hardly any rigorous results on the topic, despite a large number of numerical studies confirming the widespread existence of SNA in quasiperiodically forced systems. Here, we prove the existence of SNA in quasiperiodically forced circle maps under rather general conditions, which can be stated in terms of $${{\mathcal C}^1}$$ -estimates. As a consequence, we obtain the existence of SNA for parameter sets of positive measure in suitable parameter families. These SNA carry the unique physical measure of the system, which determines the behaviour of Lebesgue-almost all initial conditions. Finally, we show that the dynamics are minimal in the considered situations. The results apply in particular to a forced version of the Arnold circle map. For this example, we also describe how the first Arnold tongue collapses and looses its regularity due to the presence of strange non-chaotic attractors and a related unbounded mean motion property.

Using Ulam's method to calculate entropy and other dynamical invariants

Using a special form of Ulam's method, we estimate the measure-theoretic entropy of a triple , where M is a smooth manifold, T is a uniformly hyperbolic map, and is the unique physical measure of T. With a few additional calculations, we also obtain numerical estimates of (i) the physical measure , (ii) the Lyapunov exponents of T with respect to , (iii) the rate of decay of correlations for with respect to test functions, and (iv) the rate of escape (for repellors). Four main situations are considered: T is everywhere expanding, T is everywhere hyperbolic (Anosov), T is hyperbolic on an attracting invariant set (axiom A attractor), and T is hyperbolic on a non-attracting invariant set (axiom A non-attractor/repellor).