Higher-dimensional Bruckner–Garg type theorem

Abstract. A set A ⊂ C[0, 1] is shy or Haar null (in the sense of Christensen) if there exists a Borel set B ⊂ C[0, 1] and a Borel probability measure µ on C[0, 1] such that A ⊂ B and µ (B + f ) = 0 for all f ∈ C[0, 1]. The complement of a shy set is called a prevalent set. We say that a set is Haar ambivalent if it is neither shy nor prevalent.The main goal of the paper is to answer the following question: What can we say about the topological properties of the level sets of the prevalent/nonshy many f ∈ C[0, 1]?The classical Bruckner-Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) f ∈ C[0, 1] from the topological point of view. We prove that the functions f ∈ C[0, 1] for which the same characterization holds form a Haar ambivalent set.In an earlier paper we proved that the functions f ∈ C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set in C[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions f ∈ C[0, 1] for which positively many level sets with respect to the occupation measure λ • f −1 are not perfect form a Haar ambivalent set in C[0, 1].We show that for the prevalent f ∈ C[0, 1] for the generic y ∈ f ([0, 1]) the level set f −1 (y) is perfect.Finally, we answer a question of Darji and White by showing that the set of functions f ∈ C[0, 1] for which there exists a perfect set P f ⊂ [0, 1] such that f ′ (x) = ∞ for all x ∈ P f is Haar ambivalent.

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“…What can we say from the point of view of prevalence? For the generic version of this last question, see, for example, [9].…”

Section: Open Problemsmentioning

confidence: 99%

Bruckner–Garg-Type Results with Respect to Haar Null Sets inC[0, 1]

Balka

Darji

Elekes

2016

Proceedings of the Edinburgh Mathematical Society

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“…Hence it follows from Lemma 3.8 that if a set R ⊆ C t (X) is residual, then P −1 (R) is a residual subset of C n (X). Kato [K,Theorem 4.6] proved that if X is a compact metric space with dim T X = n, then for a residual set of functions f ∈ C n (X), we have (1) dim T f (X) = n; (2) There exists…”

Section: Lemma 32 the Setmentioning

confidence: 99%

Dimension and measure for generic continuous images

Balka¹,

Farkas²,

Fraser³

et al. 2013

Ann. Acad. Sci. Fenn. Math.

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Abstract. We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, X, into R n . The key question is 'what is the generic dimension of f (X)?' and we consider two different approaches to answering it: Baire category and prevalence.In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, n, but find that the behaviour of the Hausdorff, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of n and the topological dimension of X. We also study the typical Hausdorff and packing measures of f (X) and, in particular, give necessary and sufficient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of f (X) and give some simple applications concerning the prevalent dimensions of graphs of real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.

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“…The following theorem characterizes the infinite fibers of a generic f ∈ C(K, R n ). Its first part follows from a result of Kato [18,Theorem 4.6], while the second part is an easy corollary of Theorem 4.12.…”

Section: Introductionmentioning

confidence: 98%

Dimensions of fibers of generic continuous maps

Balka

2017

Monatsh Math

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Abstract. In an earlier paper Buczolich, Elekes, and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space K by introducing the notion of topological Hausdorff dimension. Later on, the author extended the theory for maps from K to R n . The main goal of this paper is to generalize the relevant results for topological and packing dimensions and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension.

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Higher-dimensional Bruckner–Garg type theorem

Cited by 4 publications

References 7 publications

Bruckner–Garg-Type Results with Respect to Haar Null Sets inC[0, 1]

Bruckner–Garg-Type Results with Respect to Haar Null Sets inC[0, 1]

Dimension and measure for generic continuous images

Dimensions of fibers of generic continuous maps

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