Bony attractors

Kudryashov, Yury

doi:10.1007/s10688-010-0028-8

Cited by 24 publications

(42 citation statements)

References 2 publications

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“…They looked at diffeomorphisms that are mapping a bounded interval strictly inside itself. They showed that so called bony graphs (after Kudryashov, see [6]) arise as attractors: these attractors are the union of a measurable graph and a zero measure set of intervals inside fibers (the bones).…”

Section: Introductionmentioning

confidence: 99%

Skew products of interval maps over subshifts

Gharaei

Homburg

2016

Journal of Difference Equations and Applications

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We treat step skew products over transitive subshifts of finite type with interval fibers. The fiber maps are diffeomorphisms on the interval; we assume that the end points of the interval are fixed under the fiber maps. Our paper thus extends work by V. Kleptsyn and D. Volk who treated step skew products where the fiber maps map the interval strictly inside itself. We clarify the dynamics for an open and dense subset of such skew products. In particular we prove existence of a finite collection of disjoint attracting invariant graphs. These graphs are contained in disjoint areas in the phase space called trapping strips. Trapping strips are either disjoint from the end points of the interval (internal trapping strips) or they are bounded by an end point (border trapping strips). The attracting graphs in these different trapping strips have different properties. ARTICLE HISTORY

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Section: Introductionmentioning

confidence: 99%

Skew products of interval maps over subshifts

Gharaei

Homburg

2016

Journal of Difference Equations and Applications

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“…The proof of Proposition 3 uses a method due to Kudryashov [6]. Proposition 3 directly implies Proposition 2.…”

Section: Proposition 3 Let V P Be the Same Neighborhood As At The Bementioning

confidence: 92%

“…In [6], Kudryashov studied attractors of skew products over Σ 2 with fiber the interval for which the theorem stated above is true. These attractors were called bony.…”

Section: Theoremmentioning

confidence: 99%

“…If relation (14) is true, then the attractor A M (F ) is Lyapunov unstable because of the hyperbolicity of the map f 1 | R 1 . On the other hand, the maps with properties 1-5 and 6 are dense in the space of skew products of the form (6). The validity of (14) would give us an example of a set dense in some open subset of the space of skew products over the Bernoulli shift such that any map from this set has Lyapunov unstable Milnor attractor.…”

Section: Problemsmentioning

confidence: 99%

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Multidimensional bony attractors

Ilyashenko

2012

Funct Anal Its Appl

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In this paper we study attractors of skew products, for which the following dichotomy is ascertained. These attractors either are not asymptotically stable or possess the following two surprising properties. The intersection of the attractor with some invariant submanifold does not coincide with the attractor of the restriction of the skew product to this submanifold but contains this restriction as a proper subset. Moreover, this intersection is thick on the submanifold, that is, both the intersection and its complement have positive relative measure. Such an intersection is called a bone, and the attractor itself is said to be bony. These attractors are studied in the space of skew products. They have the important property that, on some open subset of the space of skew products, the set of maps with such attractors is, in a certain sense, prevalent, i.e., "big." It seems plausible that attractors with such properties also form a prevalent subset in an open subset of the space of diffeomorphisms.

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“…In the next example, we consider the underlying IFS of the bony attractor in [14]. Consider the IFS(f 1 , f 2 ) whose maps f 1 and f 2 are defined on [0, 1] as follows (see Figure 3):…”

Section: Examplesmentioning

confidence: 99%

Non-hyperbolic iterated function systems: semifractals and the chaos game

Díaz

Matias

2020

Fund. Math.

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We consider iterated functions systems (IFS) on compact metric spaces and introduce the concept of target sets. Such sets have very rich dynamical properties and play a similar role as semifractals introduced by Lasota and Myjak do for regular IFSs. We study sufficient conditions which guarantee that the closure of the target set is a local attractor for the IFS. As a corollary, we establish necessary and sufficient conditions for the IFS having a global attractor. We give an example of a non-regular IFS whose target set is nonempty, showing that our approach gives rise to a "new class" of semifractals. Finally, we show that random orbits generated by IFSs draws target sets that are "stable".2000 Mathematics Subject Classification. 37C70, 28A80, 47H10.

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Bony attractors

Cited by 24 publications

References 2 publications

Skew products of interval maps over subshifts

Skew products of interval maps over subshifts

Multidimensional bony attractors

Non-hyperbolic iterated function systems: semifractals and the chaos game

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