Titlebook: Laws of Chaos; Invariant Measures a Abraham Boyarsky,Paweł Góra Book 19971st edition Springer Science+Business Media New York 1997 Generato

LARK

外科医生

Introduction,shown in Figure 1.1.1. If τ is expanding on each piece, i.e., ∣τ′(.)∣ > 1, we shall prove that τ behaves chaotically in a manner that can be described by an absolutely continuous invariant measure (acim). The theory and applications of these measures are the subjects of this book.

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Absolutely Continuous Invariant Measures,ions having an acim were known to Ulam and von Neumann [Ulam and von Neumann, 1940]. Rényi [Rényi, 1957] was the first one to define a class of transformations that have an acim. His key idea of using distortion estimates has been used in more general proofs [Adler and Flatto, 1991].

向前变椭圆

Other Existence Results,lklore Theorem which established the existence of absolutely continuous invariant measure for Markov transformations. Inspired by number theoretical questions Rényi [1957] proved the first version of this theorem for piecewise onto transformations. We follow closely the development in [Adler and Flatto, 1991].

Asperity

Spectral Decomposition of the Frobenius-Perron Operator,or. In this chapter we will study the complete set of eigenfunctions of the Frobenius-Perron operator. To do this we will need an important result from functional analysis ([Ionescu-Tulcea and Marinescu, 1950]).

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Properties of Absolutely Continuous Invariant Measures,val. Chapter 7 gave information on how a transformation decomposes the underlying space into sets each of which supports an acim. In this chapter we present properties of the absolutely continuous invariant measures themselves by studying the densities of these measures.

jaunty

故意

Compactness Theorem and Approximation of Invariant Densities,)}, it is important to be able to compute it. Unfortunately, solving the functional equation ... = . explicitly for . is possible only in very simple cases. In this chapter we investigate various procedures for approximating .*

著名

Stability of Invariant Measures, with the question of stability of properties of chaotic dynamical systems under such perturbations. Since the existence of an acim is an important property describing asymptotic statistical behavior, it is of interest to discuss the stability of an acim for a system that possesses one.

CLAP

The Inverse Problem for the Frobenius-Perron Equation,own. From the distribution of data points one can construct a probability density function on .. The inverse problem for the Frobenius-Perron equation involves determining a point transformation τ : .→. such that the dynamical system ..= τ(..) has . as its unique invariant probability density function.