### Citations

1298 | Geometric Measure Theory - Federer - 1969 |

906 |
Geometry of Sets and Measures in Euclidean Spaces. Cambridge studies in advanced mathematics 44
- Mattila
- 1995
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Citation Context ...se the standard notations dimH,dimM,dimM, and dimA to denote the Hausdorff, lower Minkowski, upper Minkowski and Assouad dimension, respectively. For basic properties of these dimensions, we refer to =-=[10, 25]-=- and especially [24] for the Assouad dimension. The paper is organized as follows. In Section 1, we present the basic definitions. In Sections 2 and 3 we study locally rich sets in metric spaces and E... |

891 |
Metric structures for Riemannian and non-Riemannian spaces
- Gromov
- 1999
(Show Context)
Citation Context ...s. To finish the paper, we consider other ways to zoom sets. 1.1. Tangents of a metric space. The distance of different metric spaces is measured by the Gromov-Hausdorff distance. For references, see =-=[1, 17, 19]-=- or [20]. The idea is that the distance of metric spaces is measured by the infimum of the Hausdorff distances of their isometric images in l∞, the space of bounded real valued sequences. This is poss... |

819 |
Fractal geometry
- Falconer
- 1990
(Show Context)
Citation Context ...se the standard notations dimH,dimM,dimM, and dimA to denote the Hausdorff, lower Minkowski, upper Minkowski and Assouad dimension, respectively. For basic properties of these dimensions, we refer to =-=[10, 25]-=- and especially [24] for the Assouad dimension. The paper is organized as follows. In Section 1, we present the basic definitions. In Sections 2 and 3 we study locally rich sets in metric spaces and E... |

228 | Lectures on analysis on metric spaces, Universitext - Heinonen |

145 | Théorie des Opérations Linéaires - Banach - 1932 |

39 |
Assouad dimension: Antifractal metrization, porous sets, and homogeneous measures
- Luukkainen
- 1998
(Show Context)
Citation Context ...s dimH,dimM,dimM, and dimA to denote the Hausdorff, lower Minkowski, upper Minkowski and Assouad dimension, respectively. For basic properties of these dimensions, we refer to [10, 25] and especially =-=[24]-=- for the Assouad dimension. The paper is organized as follows. In Section 1, we present the basic definitions. In Sections 2 and 3 we study locally rich sets in metric spaces and Euclidean spaces resp... |

24 |
Ergodic fractal measures and dimension conservation. Ergodic Theory Dynam
- Furstenberg
(Show Context)
Citation Context ...f f by Buczolich and Ráti [9]. The dynamics of the “zooming in” operation or “scenery flow” has also been studied. For example, [5, 6, 7] study the scenery flow of Cantor and Julia sets. Furstenberg =-=[16]-=- studied the distribution of tangent measures in euclidean unit cube and this idea has been further developed in the study of typical tangent measure distributions, see [21, 22]. We study compact metr... |

17 | Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets. Ergodic Theory Dynam
- Bedford, Fisher
- 1997
(Show Context)
Citation Context ... This work was later extended to the local maximum and minimum points of f by Buczolich and Ráti [9]. The dynamics of the “zooming in” operation or “scenery flow” has also been studied. For example, =-=[5, 6, 7]-=- study the scenery flow of Cantor and Julia sets. Furstenberg [16] studied the distribution of tangent measures in euclidean unit cube and this idea has been further developed in the study of typical ... |

17 |
2003) Geometric embeddings of metric spaces
- Heinonen
(Show Context)
Citation Context ...s. To finish the paper, we consider other ways to zoom sets. 1.1. Tangents of a metric space. The distance of different metric spaces is measured by the Gromov-Hausdorff distance. For references, see =-=[1, 17, 19]-=- or [20]. The idea is that the distance of metric spaces is measured by the infimum of the Hausdorff distances of their isometric images in l∞, the space of bounded real valued sequences. This is poss... |

16 |
Les dimensions d’un ensemble abstrait
- Fréchet
- 1910
(Show Context)
Citation Context ... short. We say that a compact metric space 1This theorem is sometimes represented as a consequence of the Kuratowski embedding theorem. We learned from [20, Notes to chapter 3] that actually Fréchet =-=[15]-=- proved his theorem already in 1909, while Kuratowski’s paper [23] appeared in 1935. LOCALLY RICH COMPACT SETS 3 Y is a tangent of X at x if there exists a sequence ri ↘ 0 so that Tx,ri(X) dGH−→ Y as ... |

16 |
Sobolev Spaces on Metric Measure Spaces : an Approach Based on Upper Gradients
- Heinonen, Koskela, et al.
(Show Context)
Citation Context ...e paper, we consider other ways to zoom sets. 1.1. Tangents of a metric space. The distance of different metric spaces is measured by the Gromov-Hausdorff distance. For references, see [1, 17, 19] or =-=[20]-=-. The idea is that the distance of metric spaces is measured by the infimum of the Hausdorff distances of their isometric images in l∞, the space of bounded real valued sequences. This is possible by ... |

15 |
Topics on analysis in metric spaces, volume 25
- Ambrosio, Tilli
- 2004
(Show Context)
Citation Context ...using a different method. Buczolich studied the (micro-)tangent sets of continuous functions and obtained results in a very similar fashion. He proved that the graph of typical continuous function on =-=[0,1]-=- has a graph of any other continuous function (on [-1,1] for which f(0) = 0) as a tangent set at (x, f(x)) for Lebesgue almost all x ∈ (0, 1), see [8, Theorem 5]. This work was later extended to the l... |

12 |
On the magnification of Cantor sets and their limit models
- Bedford, Fisher
- 1996
(Show Context)
Citation Context ... This work was later extended to the local maximum and minimum points of f by Buczolich and Ráti [9]. The dynamics of the “zooming in” operation or “scenery flow” has also been studied. For example, =-=[5, 6, 7]-=- study the scenery flow of Cantor and Julia sets. Furstenberg [16] studied the distribution of tangent measures in euclidean unit cube and this idea has been further developed in the study of typical ... |

10 | Null sets for doubling and dyadic doubling measures
- Wu
- 1993
(Show Context)
Citation Context ...ing measures on X. Let E ⊂ X, we say that E is thin if E has zero measure for all doubling measures on X. It is well known that porous set are thin, but the converse is not true, for more details see =-=[30]-=-. In the following, we only consider D([0, 1]d). It’s not hard to see that by applying the density theorem and Proposition 5.10 we have the following fact. For the convenience of reader, we show the d... |

9 | The scenery flow for hyperbolic Julia sets
- Bedford, Fisher, et al.
(Show Context)
Citation Context ... This work was later extended to the local maximum and minimum points of f by Buczolich and Ráti [9]. The dynamics of the “zooming in” operation or “scenery flow” has also been studied. For example, =-=[5, 6, 7]-=- study the scenery flow of Cantor and Julia sets. Furstenberg [16] studied the distribution of tangent measures in euclidean unit cube and this idea has been further developed in the study of typical ... |

9 |
Dynamics on fractals and fractal distributions
- Hochman
- 2010
(Show Context)
Citation Context ...nd Julia sets. Furstenberg [16] studied the distribution of tangent measures in euclidean unit cube and this idea has been further developed in the study of typical tangent measure distributions, see =-=[21, 22]-=-. We study compact metric spaces and compact sub-sets of euclidean spaces, and their tangent properties. In our results, there are no measures involved. We call a compact metric space locally rich, if... |

7 |
A local version of the Projection Theorem and other results in Geometric Measure Theory
- O’Neil
- 1994
(Show Context)
Citation Context ...owever, this is not always the case. O’Neil [28] constructed a Radon measure µ on Rd that has any other Radon measure of Rd as a tangent measure at µ almost all points. Furthermore, in his PhD thesis =-=[27]-=-, he showed that this is a typical property of Radon measures. This was later re-proved by Sahlsten [29] by using a different method. Buczolich studied the (micro-)tangent sets of continuous functions... |

6 |
A measure with a large set of tangent measures
- O’Neil
- 1995
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Citation Context ...ar structure than the original object in question. For example, a tangent set of a totally disconnected self-affine carpet can contain intervals, see [4]. However, this is not always the case. O’Neil =-=[28]-=- constructed a Radon measure µ on Rd that has any other Radon measure of Rd as a tangent measure at µ almost all points. Furthermore, in his PhD thesis [27], he showed that this is a typical property ... |

5 | Micro tangent sets of continuous functions - Buczolich |

5 | Mauldin and Mariusz Urbański. Dimensions and measures in infinite iterated function systems - Daniel - 1996 |

3 | Local geometry of fractals given by tangent measure distributions - Bandt |

3 | Tangent measures of typical measures
- Sahlsten
(Show Context)
Citation Context ...Radon measure of Rd as a tangent measure at µ almost all points. Furthermore, in his PhD thesis [27], he showed that this is a typical property of Radon measures. This was later re-proved by Sahlsten =-=[29]-=- by using a different method. Buczolich studied the (micro-)tangent sets of continuous functions and obtained results in a very similar fashion. He proved that the graph of typical continuous function... |

2 |
The net measure properties of symmetric Cantor sets and their applications
- Feng, Rao, et al.
- 1997
(Show Context)
Citation Context ...f dimension s and one can tangent out {0} at all its points. For the proof, we briefly give the construction of homogeneous Cantor sets. For more details, see e.g. [25, Chapter 4], [10, Chapter 4] or =-=[14]-=-. Let {mk}∞k=1 ⊂ N with mk ≥ 2 for all k ∈ N and {λk}∞k=1 ⊂ (0, 1) with the property mkλk < 1 for all k ∈ N. Let E0 be the unit interval [0, 1]. For interval [0, 1] the 1 th level intervals I1, · · · ... |

2 | Structure of distributions generated by the scenery flow. Preprint, available at http://arxiv.org/abs/1312.2567
- Käenmäki, Sahlsten, et al.
- 2013
(Show Context)
Citation Context ...nd Julia sets. Furstenberg [16] studied the distribution of tangent measures in euclidean unit cube and this idea has been further developed in the study of typical tangent measure distributions, see =-=[21, 22]-=-. We study compact metric spaces and compact sub-sets of euclidean spaces, and their tangent properties. In our results, there are no measures involved. We call a compact metric space locally rich, if... |

1 |
and Antti Käenmäki. Local structure of self-affine sets. Ergodic Theory Dynam
- Bandt
(Show Context)
Citation Context ...t set. These tangent objects often have more regular structure than the original object in question. For example, a tangent set of a totally disconnected self-affine carpet can contain intervals, see =-=[4]-=-. However, this is not always the case. O’Neil [28] constructed a Radon measure µ on Rd that has any other Radon measure of Rd as a tangent measure at µ almost all points. Furthermore, in his PhD thes... |

1 |
Category and dimension of compact subsets of Rn
- Feng, Wu
- 1997
(Show Context)
Citation Context ...nt E of G̃ satisfies dimME = 0. Thus a typical compact set of K has zero lower box-counting dimension. The result that typically a compact has zero lower box-counting dimension, was proved earlier in =-=[13]-=-, by a different method. 4. Locally rich infinitely generated self-similar set Here we consider the sub-space K+0 of K, where K+0 = {K ∈ K : 0 ∈ K and x1 ≥ 0 for all x ∈ K}. It is clear that K+0 ≈ K a... |

1 |
Quelques problèmes concernat les espaces métriques non séparables
- Kuratowski
- 1935
(Show Context)
Citation Context ...mes represented as a consequence of the Kuratowski embedding theorem. We learned from [20, Notes to chapter 3] that actually Fréchet [15] proved his theorem already in 1909, while Kuratowski’s paper =-=[23]-=- appeared in 1935. LOCALLY RICH COMPACT SETS 3 Y is a tangent of X at x if there exists a sequence ri ↘ 0 so that Tx,ri(X) dGH−→ Y as i → ∞. We denote the collection of all tangent spaces of X at x by... |