Ergodic Properties of Invariant Measures for Piecewise Monotonic Transformations

Hofbauer, Franz; Keller, Gerhard

doi:10.1007/978-0-387-21830-4_10

Cited by 153 publications

(182 citation statements)

References 10 publications

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“…Keller proved in [3] that for piecewise expanding transformations S the sequences {PJ/} (/ € L1) are asymptotically periodic and M. Misiurewicz obtained a somewhat analogous result [13] for a class of transformations with negative Schwarzian derivative.…”

Section: Introductionmentioning

confidence: 91%

Asymptotic Periodicity of the Iterates of Markov Operators

Lasota¹,

Li²,

Yorke³

1984

Transactions of the American Mathematical Society

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ABSTRACT. We say P: L1 -. L1 is a Markov operator if (i) Pf > 0 for / > 0 and (ii) ||P/|| = U/H if / > 0. It is shown that any Markov operator P has certain spectral decomposition if, for any / 6 ¿' with / > 0 and ||/|| = 1, Pnf -> 7 when n -> oo, where 7 is a strongly compact subset of L1. It follows from this decomposition that Pn f is asymptotically periodic for any / G Ll.Introduction.In the theory of stationary discrete time Markov processes, the sequence {Pn} of the iterates of a linear operator P: L1 -♦ L1 plays an important role. This operator is positive, bounded and generalizes the notion of the transition function [2,4]. Asymptotic behavior of {Pn} depends heavily on the spectral properties of P [12], but the construction of the spectral decomposition for P is not an easy problem. It can be solved under some additional hypothesis such as quasicompactness [14] of P or the uniform stability of {Pn} in mean [6]. In general, there is no technique available for studying the asymptotic behavior of {Pn} even if P is given by a simple explicit formula. A typical example where such a situation occurs is the statistical theory of deterministic systems. If 5 is a mapping of the unit interval into itself and / is a probability density, then the sequence {Pg }, with

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Section: Introductionmentioning

confidence: 91%

Asymptotic Periodicity of the Iterates of Markov Operators

Lasota¹,

Li²,

Yorke³

1984

Transactions of the American Mathematical Society

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“…It remains to estimate the norm of TT(1). We can choose a diagonal subsequence of {p ( J fceN which converges pointwise to TT (1) with its derivatives up to order r -2 . Hence ||TJ-(1)||,._ 2 < C. NOW using the method of the proof of remark 4.3(b) (in particular V k ) we obtain:…”

Section: B Szewcmentioning

confidence: 99%

“…Introduction A. Lasota and J. Yorke [5] started the long series of studies of ergodic properties of piecewise monotonic transformations of an interval with derivative greater than one. Advanced research in this domain is found in [1] which is written in terms of functional analysis. M. Rychlik [8] has obtained similar results for maps with a countable number of pieces of monotonicity and without the application of the strong Ionescu-Tulcea, Marinescu theorem.…”

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confidence: 99%

The Perron-Frobenius operator in spaces of smooth functions on an interval

Szewc¹

1984

Ergod. Th. Dynam. Sys.

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Abstract. The densities of invariant measures for Misiurewicz maps and LasotaYorke maps of class C are of class C r~l on certain intervals (forming the partition of an interval in case of Misiurewicz maps). For these maps the Perron-Frobenius operator has an unambiguous decomposition into the sum of projections onto eigenspaces (multiplied by the eigenvalues) and a remainder operator. The remainder operator has spectral radius less than one in certain spaces of smooth functions. Other useful references have maps with singularities where the derivative is equal to zero (for example the famous family of quadratic maps {ax(l -x)}). I recall only the papers closely related to the present research. M. Misiurewicz [6] has proved the existence and studied properties of absolutely continuous invariant measures for negative Schwarzian maps without sinks and such that the set of critical points is separated from the trajectory. W. Szlenk [10] has proved the existence of absolutely continuous measures for similar maps which are not necessarily negative Schwarzian.The third subject connected with this research is the question of smoothness of densities of invariant measures. This question for expanding maps was answered by R. Sacksteder [9] (unfortunately the paper contains an important mistake) and K. Krzyzewski [4] who proved that if an expanding map is of class C r then the density is of class C r~' .In this paper we construct the spaces C}~1 of functions of class C~l on intervals forming a partition with a weighted sup-norm.The Perron-Frobenius operator for Lasota-Yorke maps or Misiurewicz maps has an unambiguous decomposition into the sum of projections onto the eigenspaces and the remainder operator. The spectral radius of the remainder operator in the

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“…Assuming additionally that |f | > 1 on the intervals of monotonicity, Lasota and Yorke showed that there are finitely many absolutely continuous and ergodic invariant probability measures µ j (which are SRB and physical measures). Hofbauer's spectral decomposition gives [53] proved that the associated correlation functions decay exponentially fast for observables of bounded variation. More recently, the Birkhoff cone method was applied by Liverani to show the same result [69].…”

Section: Exponential Mixing For One-dimensional Piecewise Expanding Mapsmentioning

confidence: 99%

Spectrum and Statistical Properties of Chaotic Dynamics

Baladi

2001

European Congress of Mathematics

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Abstract. We present new developments on the statistical properties of chaotic dynamical systems. We concentrate on the existence of an ergodic physical (SRB) invariant measure and its mixing properties, in particular decay of its correlation functions for smooth observables. In many cases, there is a connection (via the spectrum of a Ruelle-Perron-Frobenius transfer operator) with the analytic properties of a weighted dynamical zeta function, weighted dynamical Lefschetz function, or dynamical Ruelle-Fredholm determinant, built using the periodic orbit structure of the map.

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Ergodic Properties of Invariant Measures for Piecewise Monotonic Transformations

Cited by 153 publications

References 10 publications

Asymptotic Periodicity of the Iterates of Markov Operators

Asymptotic Periodicity of the Iterates of Markov Operators

The Perron-Frobenius operator in spaces of smooth functions on an interval

Spectrum and Statistical Properties of Chaotic Dynamics

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