Continuous Invariant Measure Research Articles

Piecewise linear interval maps both expanding and contracting

The dynamics of piecewise linear interval maps is studied with two branches, one expanding and one contracting. It is proved that such a map either has a periodic attractor or it is eventually expanding. In the latter case there exists an absolutely continuous invariant measure.

Absolutely continuous invariant measures for multidimensional expanding maps

We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps with singularities. We allow maps that are discontinuous on some extremely wild sets, the shape of the discontinuities being completely ignored with our approach.

On estimating absolutely continuous invariant measures

On estimating absolutely continuous invariant measures

The exact rate of approximation in Ulam's method

This paper investigates the exact rate of convergence in Ulam's method: a well-known discretization scheme for approximating the invariant density of an absolutely continuous invariant probability measure for piecewise expanding interval maps. It is shown by example that the rate is no better than $O(\frac{\log n}{n})$, where $n$ is the number of cells in the discretization. The result is in agreement with upper estimates previously established in a number of general settings, and shows that the conjectured rate of $O(\frac{1}{n})$ cannot be obtained, even for extremely regular maps.

Weak Gibbs measures for certain non-hyperbolic systems

We study a weak Gibbs property of equilibrium states for potentials of weak bounded variation and for maps admitting indifferent periodic points. We further establish statistical properties of the weak Gibbs measures and bounds of their pointwise dimension. We apply our results to higher-dimensional maps (which are not necessarily conformal) with indifferent periodic points and show that their absolutely continuous finite invariant measures are weak Gibbs measures.

Ergodic properties of the Bolyai-Rényi expansion

We consider the ergodic properties of the Bolyai-Rényi expansion for real numbers introduced in [Bol], [Ren]. We compute the entropy of the absolutely continuous invariant probability measure μ of the associated dynamical system, resolving a conjecture of Bosma, Dajani and Kraaikamp [BDK]. We also compute the frequencies with which the various digits occur in the expansions of Lebesgue almost every number. Finally, we calculate the rate of decay of correlations with respect to μ.

Decay of random correlation functions for unimodal maps

Since the pioneering results of Jakobson and subsequent work by Benedicks-Carleson and others, it is known that quadratic maps tf a ( χ) = a − χ 2 admit a unique absolutely continuous invariant measure for a positive measure set of parameters a. For topologically mixing tf a , Young and Keller-Nowicki independently proved exponential decay of correlation functions for this a.c.i.m. and smooth observables. We consider random compositions of small perturbations tf + ω t , with tf = tf a or another unimodal map satisfying certain nonuniform hyperbolicity axioms, and ω t chosen independently and identically in [−ϵ, ϵ]. Baladi-Viana showed exponential mixing of the associated Markov chain, i.e., averaging over all random itineraries. We obtain stretched exponential bounds for the random correlation functions of Lipschitz observables for the sample measure μ ω of almost every itinerary.

Stochastic Stability¶for Contracting Lorenz Maps and Flows

In a previous work [M], we proved the existence of absolutely continuous invariant measures for contracting Lorenz-like maps, and constructed Sinai–Ruelle–Bowen measures f or the flows that generate them. Here, we prove stochastic stability for such one-dimensional maps and use this result to prove that the corresponding flows generating these maps are stochastically stable under small diffusion-type perturbations, even though, as shown by Rovella [Ro], they are persistent only in a measure theoretical sense in a parameter space. For the one-dimensional maps we also prove strong stochastic stability in the sense of Baladi and Viana[BV].

Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbb{R}$-analytic mappings of the plane

We prove that any expanding piecewise real-analytic map of a bounded region of the plane admits absolutely continuous invariant probability measures.

Piecewise smooth interval maps with non-vanishing derivative

We consider the dynamics of piecewise smooth interval maps $f$ with a nowhere vanishing derivative. We show that if $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of $f$ is expanding. In this case, $f$ admits an absolutely continuous invariant probability measure.

Multifractal formalism for some parabolic maps

We study multifractal spectra of a simple class of interval transformations with a neutral fixed point. The analyticity of their spectral curves depends on whether the absolutely continuous invariant measure is infinite or not.

Computing Invariant Densities and Metric Entropy

We present a method for accurately computing the metric entropy (or, equivalently, the Lyapunov exponent) of the absolutely continuous invariant measure μ for a piecewise analytic expanding Markov map T of the interval. We construct atomic signed measures μ M supported on periodic orbits up to period M, and prove that super-exponentially fast. We illustrate our method with several examples.

Chaotic Monte Carlo Computation: A Dynamical Effect of Random-Number Generations

Chaotic maps with absolutely continuous invariant probability measures are implemented as random-number generators for Monte Carlo computation. We observe that such Monte Carlo computation based on chaotic random-number generators yields sometimes unexpected dynamical dependency behavior which cannot be explained by usual statistical arguments. Furthermore, we find that superefficient Monte Carlo computation with O(1/N2) mean square error can be carried out as an extreme case of such dynamical dependency behavior. Here, such superefficiency sharply contrasts with the conventional Monte Carlo simulation with O(1/N) mean square error. By deriving a necessary and sufficient condition for the superefficiency, it is shown that such high-performance Monte Carlo simulations can be carried out only if there exists a strong correlation with chaotic dynamical variables. Numerical calculation illustrates this dynamics dependency and the superefficiency of various chaotic Monte Carlo computations.

On the existence of ergodic continuous invariant measures for folding transformations

We present a sufficient condition for the existence of ergodic continuous invariant measures (ECIMs) for continuous folding transformations. We apply this result to prove the existence of ECIMs for general set expanding Markov transformations and $n$-dimensional tent maps.

SRB measures for non-hyperbolic systems with multidimensional expansion

We construct ergodic absolutely continuous invariant probability measures for an open class of non-hyperbolic surface maps introduced by Viana (1997), who showed that they exhibit two positive Lyapunov exponents at almost every point. Our approach involves an inducing procedure, based on the notion of hyperbolic time that we introduce here, and contains a theorem of existence of absolutely continuous invariant measures for multidimensional piecewise expanding maps with countably many domains of smoothness.

Absolutely Continuous Invariant Measures¶for Piecewise Real-Analytic Expanding Maps¶on the Plane

We prove the existence of absolutely continuous invariant measures for piecewise real-analytic expanding maps on bounded regions in the plane.

Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions

In this paper, by using a trace theorem in the theory of functions of bounded variation, we prove the existence of absolutely continuous invariant measures for a class of piecewise expanding mappings of general bounded domains in any dimension.

ABSOLUTELY CONTINUOUS INVARIANT MEASURES FOR GENERIC MULTI-DIMENSIONAL PIECEWISE AFFINE EXPANDING MAPS

By a well-known result of Lasota and Yorke, any self-map f of the interval which is piecewise smooth and uniformly expanding, i.e. such that inf |f′|>1, admits absolutely continuous invariant probability measures (or a.c.i.m.'s for short). The generalization of this statement to higher dimension remains an open problem. Currently known results only apply to "sufficiently expanding maps". Here we present a different approach which can deal with almost all piecewise expanding maps. Here, we consider both continuous and discontinuous piecewise affine expanding maps.

ON DYNAMICAL MODEL OF DRILLING WELLS

A family of one-dimensional mappings and their stochastic perturbations, related to the determining of cogged bits efficiency, is studied. A sufficient condition for the existence of absolutely continuous invariant probability measure is given, and the weak mixing property of this measure is established. We formulate and discuss the results concerning the stochastic stability of the invariant measure and its continuous dependence on the dimensionless parameter of the model. Some new problems are also outlined.

A finite element method for the Frobenius–Perron operator equation

We present a finite element framework to numerically solve the absolutely continuous invariant measure associated with a non-singular transformation. Both convergence analysis and error estimates of the method are given for a class of mappings in R n .