Invariant Measures of Set-Valued Maps

Artstein, Zvi

doi:10.1006/jmaa.2000.7095

Cited by 15 publications

(4 citation statements)

References 14 publications

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“…Due to the nonuniqueness of solutions this concept may be very restrictive for differential inclusion in general, indeed one may find different approaches to work with a measure preserving differential inclusions (e.g. [1] and the references therein). However, as we shall see, the definition of measure preserving (5) provides interesting results for Filippov systems.…”

Section: Differential Inclusionmentioning

confidence: 99%

A note on invariant measures for Filippov systems

Novaes,

Varão

2017

Preprint

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We are interested in Filippov systems which preserve a probability measure on a compact manifold. Using the formalism coming from the theory of differential inclusions, we define a measure to be invariant for a Filippov system as the natural analogous definition of invariant measure for flows. Our first main result states that if a differential inclusion admits an invariant probability measure, this measure does not see the trajectories where there is a break of uniqueness. Our second main result provides a necessary and sufficient condition in order to exist an invariant probability measure preserved by a Filippov system. Our third main result concerns Filippov systems which preserve a probability measure equivalent to the volume measure. As a corollary the volume preserving Filippov systems are the refractive ones. Then, in light of our previous results, we analyze the existence of invariant measures for many examples.

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Section: Differential Inclusionmentioning

confidence: 99%

A note on invariant measures for Filippov systems

Novaes,

Varão

2017

Preprint

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“…In the latter case one may employ notions for invariance for set-valued maps, as defined, e.g., in Miller and Akin [12] or in Artstein [2]. In Example 3.7 such considerations would resolve the invariance issue, and produce an ergodic pair, namely the one generated by the fixed point T (0) = 0.…”

Section: Proof It Is Enough To Prove Thatmentioning

confidence: 99%

Ergodicity and mixing via Young measures

Artstein

Grinfeld²

2002

Ergod. Th. Dynam. Sys.

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Connections are established between mixing or ergodic properties of maps on the one hand, and the convergence of the iterates of the map, or of the empirical measures of the iterates, to a constant measure-valued map, on the other. The uniqueness of an absolutely continuous ergodic measure can also be verified via the convergence. The technique helps to identify ergodic and mixing pairs and verify the uniqueness in specific examples

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“…Our work contributes to the abstract theory of set-valued dynamical systems dating back to the 1960s. Early contributions were motivated mainly by control theory [Rox65,Klo78], and later developments include stability and attractor theory [Ara00, GK01, Grü02, KMR11, McG92, Rox97], Morse decompositions [BBS,Li07,McG92] and ergodic theory [Art00] Our results build upon initial piloting studies concerning bifurcations in random dynamical systems with bounded noise [BHY, CGK08, CHK10, HY06, HY10, ZH07, ZH08] and control systems [CK03, CMKS08, CW09, Gay04, Gay05]. In particular, Theorem 1.2 unifies and generalises observations in [BHY,HY06,ZH07] to higher dimensions and non-invertible (set-valued) systems, while the bifurcation analysis in terms of Morse-like decompositions is a novel perspective.…”

Section: Introductionmentioning

confidence: 99%

Topological bifurcations of minimal invariant sets for set-valued dynamical systems

Lamb

Rasmussen

Rodrigues

2015

Proc. Amer. Math. Soc.

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Abstract. We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are naturally satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous appearances. We also characterise these transitions by properties of Morse-like decompositions.

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Invariant Measures of Set-Valued Maps

Cited by 15 publications

References 14 publications

A note on invariant measures for Filippov systems

A note on invariant measures for Filippov systems

Ergodicity and mixing via Young measures

Topological bifurcations of minimal invariant sets for set-valued dynamical systems

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