Continuous Invariant Measure Research Articles

Positive Lyapunov exponents in families of one dimensional dynamical systems

We give a generalization of Jakobson's theorem [Ja] and a precise estimate for the density of parameters corresponding to systems with absolutely continuous invariant measures.

CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS

Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

The Fibonacci Unimodal Map

This paper will study topological, geometrical and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan. It turns out that the situation can be understood completely and is of quite regular nature. In particular, any Fibonacci map (with negative Schwarzian and non-degenerate critical point) has an absolutely continuous invariant measure (so, we deal with a ``regular'' type of chaotic dynamics). It turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the Feigenbaum map: its Hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter.

Networks and dynamical systems

A new approach to the problem of classification of (deflected) random walks in or Markovian models for queueing networks with identical customers is introduced. It is based on the analysis of the intrinsic dynamical system associated with the random walk. Earlier results for small dimensions are presented from this novel point of view. We give proofs of new results for higher dimensions related to the existence of a continuous invariant measure for the underlying dynamical system. Two constants are shown to be important: the free energy M < 0 corresponds to ergodicity, the Lyapounov exponent L < 0 defines recurrence. General conjectures, examples, unsolved problems and surprising connections with ergodic theory, classical dynamical systems and their random perturbations are largely presented. A useful notion naturally arises, the so-called scaled random perturbation of a dynamical system.

Networks and dynamical systems

A new approach to the problem of classification of (deflected) random walks inor Markovian models for queueing networks with identical customers is introduced. It is based on the analysis of the intrinsic dynamical system associated with the random walk. Earlier results for small dimensions are presented from this novel point of view. We give proofs of new results for higher dimensions related to the existence of a continuous invariant measure for the underlying dynamical system. Two constants are shown to be important: the free energyM< 0 corresponds to ergodicity, the Lyapounov exponentL< 0 defines recurrence. General conjectures, examples, unsolved problems and surprising connections with ergodic theory, classical dynamical systems and their random perturbations are largely presented. A useful notion naturally arises, the so-called scaled random perturbation of a dynamical system.

A Compactness Theorem for Approximating the Invariant Densities of Higher Dimensional Transformations

Let τ be a Jablonski transformation from the n-dimensional unit cube U into itself which has a unique absolutely continuous invariant measure with density function ƒ. Let T denote a family of transformations which approximate τ on finer and finer partitions. The main result of this paper is a compactness theorem on the densities associated with T which allows us to prove that the invariant densities associated with the transformations in T converge weakly to ƒ.

Some dynamical properties of S-unimodal maps

We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet–Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron–Frobenius operator, 4) the density of the absolutely continuous invariant measure.

The Fibonacci unimodal map

The recurrence of the critical orbit appeared in the work of Branner and Hubbard on complex cubic polynomials [BH, §12] and in Yoccoz's work [Yl, Y2] on quadratic ones, as the worst pattern of recurrence. On the other hand, a real quadratic map was suggested by Hofbauer and Keller [HK] as a possible candidate for a map having a attractor (that is, a set A which is the w-limit set for Lebesgue almost every orbit but is strictly smaller than the w-limit set for a generic orbit). The w-limit set of the critical orbit in [HK] possesses all known topological properties of wild attractors (compare [BL2]). In fact, we will see below that the quadratic map does not have a wild attractor; however, the corresponding question for a map with a degenerate critical point remains open. Actually, the first indication of the map appeared in the numerical work of Tsuda [T], related to the Belousova-Zhabotinskii reaction, and also in numerical work of Shibayama [Sh] (more precisely, they studied the sequence of Fibonacci bifurcations creating the map). This paper will study topological, geometrical, and measure-theoretical properties of the real map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan [S]. It turns out that the situation can be understood completely and is of quite nature. In particular, any map (with negative Schwarzian and nondegenerate critical point) has an absolutely continuous invariant measure (so, we deal with a regular type of chaotic dynamics). It turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the Feigenbaum map: its Hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter. Branner and Hubbard introduce the concept of a in order to describe recurrence of critical orbits. Their Fibonacci tableau is a basic example, which corresponds to one particularly close and pattern of recurrence. If a complex quadratic map z 1-+ z2 + c realizes this tableau, then the

Modelling and simulating higher dimensional chaotic data

We consider time series data whose estimated Lyapunov exponents are positive. This data is represented by a probability density function on a suitably fine partition of the phase space. An algorithm is presented for constructing a transformation τ which has a unique absolutely continuous invariant measure whose density function is ƒ and which has the same Lyapunov exponents as the original time series. An example is given.

Regularity and other properties of absolutely continuous invariant measures for the quadratic family

In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved.

Error estimates of the Markov finite approximation of the Frobenius-Perron operator

IN THE practical application of the modern ergodic theory, one of the main problems is related to the computation of the absolutely continuous invariant measures for nonsingular transformations on measure spaces (see Lasota and Mackey [l]). A simple example is constructed in Li [2] to show that the round-off error can completely dominate the calculation if one is to compute the invariant measure directly using a sequence of Cesaro sums. An alternative is to compute the invariant density function for the Frobenius-Perron operator P,: L’(0, 1) + L’(0, 1) associated with the nonsingular measurable transformations S: [0, 11 + [0, 11. The Frobenius-Perron operator P, is defined by

Decay of correlations for certain quadratic maps

We prove exponential decay of correlations for (f, μ), wheref belongs in a positive measure set of quadratic maps of the interval and μ is its absolutely continuous invariant measure. These results generalize to other interval maps.

Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

AbstractWe study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

On the approximation of invariant measures

Given a discrete dynamical system defined by the map τ:X →X, the density of the absolutely continuous (a.c.) invariant measure (if it exists) is the fixed point of the Frobenius-Perron operator defined on L1(X). Ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator. T. Y. Li proved the convergence of the scheme for expanding maps of the interval. G. Keller and M. Blank extended this result to piecewise expanding maps of the cube in ℝn. We show convergence of a variation of Ulam's scheme for maps of the cube for which the Frobenius-Perron operator is quasicompact. We also give sufficient conditions onτ for the existence of a unique fixed point of the matrix approximation, and if the fixed point of the operator is a function of bounded variation, we estimate the convergence rate.

Existence of absolutely continuous invariant measures for higher-dimensional random maps

A random map is a discrete-time dynamical system in which one of a number of transformations is selected randomly and applied. The main result of this paper provides a sufficient condition of the existence of an absolutely continuous invariant measure for random maps on . Such random maps have application to the fractals that arise from iterated-function systems

Graph-theoretic bound on number of A.C.I.M. for random transformation

Upper and lower bounds are given for the number of ergodic absolutely continuous invariant measures for a general random map. They are based on a communication graph describing the interaction of deterministic transformations used to define the random one. In particular, the case of piecewise expanding transformations is discussed.

Approximating measures invariant under higher-dimensional chaotic transformations

Let τ be a Jablonski transformation from the n-dimensional unit cube into itself. We present a method for approximating the absolutely continuous invariant measures by means of approximating the Frobenius-Perron operator by finite-dimensional operators. This proves an n-dimensional version of a conjecture by Ulam and generalizes the one-dimensional results of T. Y. Li.

On the number of invariant measures for higher-dimensional chaotic transformations

LetS be a bounded region inR N and letP={S l } i=1 m be a partition ofS into a finite number of closed subsets having piecewiseC2 boundaries of finite (N−1)-dimensional measure. Let τ:S→S be piecewiseC2 onP and expanding in the sense that there exists 0<σ<1 such that for anyi=1,2,...,m, ∥DT i −1∥<σ, whereDT i −1 is the derivative matrix ofT i −1 and ∥·∥ is the Euclidean matrix norm. We prove that for some classes of such mappings, for example, Jabtonski transformations or convexity-preserving transformations, the number of crossing points constitutes a bound for the number of ergodic absolutely continuous τ-invariant measures. We give examples showing that in general the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we show that it is possible to construct piecewise expandingC2 transformations on a fixed partition with a finite number of elements but which have an arbitrarily large number of ergodic, absolutely continuous invariant measures.

Noninvertible transformations admitting no absolutely continuous 𝜎-finite invariant measure

We study a family of n n -to-1 conservative ergodic endomorphisms which we will show to admit no σ \sigma -finite absolutely continuous invariant measure. We exhibit recurrent measures for these transformations and study their ratio sets; the examples can be realized as C ∞ {C^\infty } endomorphisms of the 2 2 -torus.

Absolutely Continuous Invariant Measures for Expansive Rational Maps with Rationally Indifferent Periodic Points

Absolutely Continuous Invariant Measures for Expansive Rational Maps with Rationally Indifferent Periodic Points