MAARIT J ¨ARVENP ¨A ¨A
We discuss dimensional properties of visible parts of deterministic and random sets in Euclidean spaces. Special emphasis is given to fractal percolation. This talk is based on a joint work with I. Arhosalo, E. J¨arvenp¨a¨a and P. Shmerkin.
AMS 2000 Subject Classification: 28A80, 28A78.
Key words: visible part, fractal percolation, Hausdorff and box counting dimen- sions.
1. VISIBLE PARTS
Letnand mbe integers with 0< m < n. The visible part of a compact set A⊂Rn from an affinem-dimensional subspaceK of Rn consists of those pointsx∈Awhere one first hits the setAwhen looking perpendicularly from K. More precisely:
Definition 1.1. LetA⊂Rnbe compact and letK be an affinem-dimen- sional subspace not meeting A. The visible part of A fromK is
VK(A) ={x∈A|[x, PK(x)]∩A={x}},
wherePK(x) is the orthogonal projection ofxonto the affine subspaceK and [x, PK(x)] is the line segment joiningx toPK(x).
Visibility has been used to study star-like sets in convex analysis (see e.g. [2], [3], [8]). There exists also a measure theoretic definition for visibility (see [6] and [16]).
We address the question of how dimension of visible parts depends on that of the original set. As Example 1.2 indicates, only “almost all” type of results are possible; there are always exceptional directions. Note that there exist natural Radon measures Γn,m and γn,m on the space A(n, m) of affine m-dimensional subspaces of Rn and on the Grassmann manifold G(n, m) of linear m-dimensional subspaces of Rn, respectively. Since in the plane every line through the origin can be parametrized by the angle which it makes with the positive x-axis, a normalized restriction of the 1-dimensional Lebesgue measure to the interval [0, π] induces the measure γ2,1. Similarly, for lines in
REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 451–459
Rn the surface measure on the sphere Sn−1 induces the measure γn,1. The case m =n−1 is also easy to handle by identifying hyperplanes with their orthogonal complements. However, in the case 2≤m≤n−2 the method for constructing the measureγn,mis more complicated. It is based on the existence of the Haar measure θn on the orthogonal group On consisting of all linear mappings g : Rn → Rn that preserve the distance. Fixing V ∈ G(n, m), we define a Radon probability measureγn,m byγn,m(A) =θn({g∈ On:gV ∈A}) for all A⊂G(n, m). It turns out that the definition ofγn,m is independent of the choice of V. For details see [17, 3.9].
The measures Γn,m and γn,m are connected by means of the equation below: for all Borel sets A⊂A(n, m) we have [17, 3.16]
Γn,m(A) = Z
Hn−m({a∈V⊥|V +a∈A}) dγn,m(V),
where Hn−m is the (n−m)-dimensional Hausdorff measure and V⊥ is the orthogonal complement of V ∈G(n, m).
We denote by dimH and dimB the Hausdorff and upper box counting dimensions, respectively. Note that dimHA ≤ dimBA for all bounded sets A⊂Rn.
Example 1.2. (a) Let A ⊂ R2 be a graph of a continuous function f : [0,1] → [0,1]. Then dimH(VL(A)) = 1 for all affine lines L which do not intersect Awith the possible exception of one direction.
Indeed, at every point of VL(A) the open cone determined by the line which is parallel to the y-axis and the line which is perpendicular to L is in the complement ofVL(A). By [17, Lemma 15.13], this implies thatVL(A) is 1- rectifiable and, therefore, dimH(VL(A))≤1. On the other hand, dimH(VL(A))
≥1 sincePL(VL(A)) contains an interval, except in the case thatf is affine and Lis perpendicular to its graph. IfLis parallel to thex-axis, thenVL(A) =A.
(b) Modifying (a), for fixed 1 ≤ s ≤ 2, consider a compact set A ⊂ R2 with dimHA = s such that dimHVL(A) = 1 for typical lines L but the visible parts from countably many exceptional lines have Hausdorff dimension s. Using a construction given by Davies and Fast [7] one obtains an example where the exceptional set is a dense Gδ-set.
E. J¨arvenp¨a¨a, M. J¨arvenp¨a¨a, MacManus and O’Neil [12] proved that if dimHA≤n−1 then
(1.1) dimHVK(A) = dimHA
for Γn,m-almost all m-dimensional affine subspaces K. On the other hand, if dimHA > n−1 then
(1.2) n−1≤dimHVK(A)
for Γn,m-almost allm-dimensional affine subspacesK. It is an open question whether the opposite inequality holds in (1.2).
It is easy to see thatn−1 is the only possible constant value for Hausdorff dimension of typical visible parts. More precisely, if for all compact sets A⊂ Rn with dimHA > n−1 there exists a constantc such that dimHVK(A) =c for almost allK, thenc=n−1. Indeed, assume to the contrary thatc > n−1 for some A ⊂Rn. Take B = A∪H, where H is a suitably chosen subset of a hyperplane which is disjoint from A. Clearly, dimHB = dimHA > n−1.
Moreover, there are E, F ⊂ A(n, m) with positive Γn,m-measures such that dimHVK(A) =cfor all K ∈E and dimHVK(A) =n−1 for all K ∈F. This gives a contradiction.
Results (1.1) and (1.2) resemble the Marstrand–Kaufman–Mattila-type projection results according to which
(1.3) dimHPV(A) = min{dimHA, m}
for γn,m-almost all m-dimensional linear subspaces V (see [15], [14], [18]).
Indeed, the methods utilized in [12] for proving (1.1) and (1.2) are based on the generalized projection formalism for parametrized families of transversal mappings due to Peres and Schlag [20]. However, the difference between (1.1), (1.2) and (1.3) is crucial in the following sense: in (1.3) the upper bound dimHPV(A) ≤ min{dimHA, m} is trivial since Hausdorff dimension cannot increase under projections while the reverse inequality of (1.2) is non-trivial.
Questions related to the open problem of whether the opposite inequality holds in (1.2) is investigated in [19] and [13]. O’Neil [19] showed that if a compact connected plane set has dimension strictly greater than one, then the Hausdorff dimension of a typical visible part is strictly less than that of the original set. The higher dimensional case is considered in [13]. Given a set A⊂Rn with dimHA > n−1, it follows from the relation between Hausdorff dimensions of sets and those of measures that there exists a non-trivial finite Radon measure µ on A with dimHµ > n−1. In [13] it is verified that the visible parts of A are typically smaller than the setA itself in the sense that they have typically µ-measure zero for any such measureµ.
Quasi-circles are examples of plane sets that have Hausdorff dimension strictly greater than one and for which the dimension of visible parts from all affine subspaces is one [12].
As mentioned earlier, it is an open question if the dimension of a visible part of a compact subset A of Rn is typically n−1 under the assumption dimHA > n−1. For fractal percolation in the plane we prove in [1] that with probability one this is true. In the next section we discuss the construction and some basic properties of fractal percolation. The last section is dedicated to explaining the crucial ideas of [1].
2. FRACTAL PERCOLATION
Fix 0 < p < 1. We construct a random compact set as follows. Let Q0 = [0,1]×[0,1] be the unit square in R2 and divide Q0 into four equal subsquares. Each of these subsquares is chosen with probabilitypand dropped with probability 1 −p, independently of each other. Denote by C1 be the collection of all chosen subsquares at the first level. For each Q ∈ C1, we continue the same process by dividing Q into four subsquares of equal size.
Again each of these subsquares is chosen with probabilitypand dropped with probability 1−p, independently of each other. The set of all chosen squares at the second level is denoted by C2. Repeating this process inductively gives the limiting random set E defined as
E=
∞
\
n=1
S{Q:Q∈ Cn}.
Throughout this paper we denote by E a random set constructed in this manner and by Ω the space of all such constructions.
We construct a probability measurePon Ω in the following natural way:
Given E ∈ Ω, let En be the union of all chosen squares of side length 2−n, that is,
En=S
{Q:Q∈ Cn}.
For a configuration E1 we give the weightpk(1−p)4−k, wherekis the number of squares in C1. Similarly, we attach weights to the configurations inside each Q ∈ C1. Continuing in this manner gives weights to each cylinder set {E ∈ Ω : En = S
Q∈FQ}. The cylinders induce the natural topology to the space Ω and the measure Pcan be defined using weights of cylinders. We skip the technical details (see [11]) since for our purposes it is sufficient to know how to calculate weights of cylinder sets. In what follows “almost surely” refers always to the probability measureP.
Note that there is a positive probability of E being empty. In fact, P(E=∅) = 1 if and only ifp≤1/4.
Conditioned on non-extinction, that is, E6=∅, we have dimHE= log(4p)/log 2 almost surely. This implies that, conditioned on non-extinction, dimHE >1 almost surely provided that p > 1/2. In particular, when considering di- mensional properties of visible parts of E in Section 3, we may restrict our consideration to the case p > 1/2. In the case 1/4 < p ≤ 1/2 we can use equation (1.1).
J.T. Chayes, L. Chayes and Durrett [5] verified that there is a critical probability 0< pc <1 such that ifp < pc then with probability one the largest connected component of E is a point, whereas the opposing sides of Q0 are
connected with positive probability provided that p≥pc. A good overview of results concerning fractal percolation can be found in [11] and [4].
We will need in Section 3 the next lemma due to K.J. Falconer and G.R.
Grimmett, when considering dimensional properties of visible parts of E.
Lemma 2.1. Assume that p > 1/2. Let P0 be the projection onto the x-axis. Then
P(P0(E) = [0,1]) =:q(p)>0 and
P(P0(E)contains an interval|E 6=∅) = 1.
Proof. See [9] and [10].
3. DIMENSIONAL PROPERTIES OF VISIBLE PARTS OF FRACTAL PERCOLATION
In this section we state the main theorem of [1] and give a sketch of the proof. Let L be any affine line. Then, conditioned on non-extinction, PL(E) contains an interval almost surely if and only if PL(E) contains an interval with positive probability.
Theorem 3.1. Let L be any affine line which does not meet the unit square. Assume that PL(E)contains an interval almost surely, conditioned on non-extinction. Then
dimH(VL(E)) = dimB(VL(E)) = 1 and 0<H1(VL(E))<∞ almost surely conditioned on non-extinction.
As an immediate consequence we have
Corollary3.2.Letp≥pc. Conditioned on non-extinction, almost surely dimH(VL(E)) = dimB(VLE)) = 1 and 0<H1(VL(E)<∞
for almost all affine lines L which do not meet the unit square.
Proof. Since p ≥ pc the opposing sides are connected with positive probability by [5]. This in turn implies that, conditioned on non-extinction, almost surely there is a construction square Qcontaining a crossing from left to right. SincePL(E∩Q) contains an interval for allL, the claim follows from Theorem 3.1 and Fubini’s theorem. For measurability arguments see [1].
Remark 3.3. (a) According to results by M. Rams and K. Simon (in preparation), the assumption of Theorem 3.1 is valid for all affine linesLunder the condition that dimE >1, or equivalently,p >1/2. Theorem 3.1 combined
with (1.1) gives then a complete answer to the question of determining the dimensions of visible parts of E.
(b) A straightforward consequence of Lemma 2.1 is that Theorem 3.1 is valid for all lines which are parallel to the x-axis and do not meet the unit square provided that p >1/2.
We proceed by presenting the proof of Theorem 3.1 in the case whereL is an affine line parallel to the x-axis. Even though the proof in the general case is quite technical and requires auxiliary lemmas the main ideas can be clearly illustrated in the following simplification.
Proof of Theorem 3.1 for lines L that are parallel to the x-axis. Let P0 be the orthogonal projection onto the x-axis. By Lemma 2.1, P0(VL(E)) = P0(E) has positive H1-measure almost surely conditioned on non-extinction.
Since Hausdorff measure does not increase under projections, this implies that H1(VL(E)) > 0 and dimH(VL(E)) ≥ 1 almost surely conditioned on non- extinction. Hence, it is sufficient to prove that
H1(VL(E))<∞ and dimB(VL(E))≤1 almost surely.
Let Q ∈ Cn for some n. We say that Q is a block if one cannot see through E∩Qfrom thex-axis, more precisely, ifP0(E∩Q) =P0(Q). Other- wise, we say that Q is a window. By independence and Lemma 2.1, any Q ∈ Cn has the same probability q :=q(p) of being a block. Moreover, if Q1 and Q2 are disjoint chosen squares, then “Q1 is a block” and “Q2 is a block”
are independent events. In particular, this applies to chosen squares of the same level.
For each j∈ {0, . . . ,2n−1}, we denote by Cn,j the set of those squares in Cn which belong to thejth column inside Q0, that is,
Cn,j ={Q∈ Cn:Q= [j2−n,(j+ 1)2−n]×
×[i2−n,(i+ 1)2−n] for somei∈ {0, . . . ,2n−1}}.
Let Xn(j) =|Cn,j|, where|Cn,j|is the number of squares in Cn,j. Define Cn,j∗ ={Q∈ Cn,j : anyQ0 ∈ Cn,j below Qis a window},
and set Yn(j) = |Cn,j∗ |. Note that if all squares in Cn,j are windows, then Yn(j) = Xn(j). Otherwise, Yn(j) is one plus the number of windows above [j2−n,(j+ 1)2−n] which are closer to thex-axis than any block.
For the purpose of estimating the upper box counting dimension of the visible part VL(E) from above, the following observation is crucial: for alln
(3.1) VL(E)⊂
2n−1
[
j=0
S{Q:Q∈ Cn,j∗ }.
Indeed, since VL(E)⊂E any point z∈VL(E) belongs to someQ∈ Cn. Fix j such that Q∈ Cn,j. IfQ /∈ Cn,j∗ then there is a squareQ0 ∈ Cn,j belowQwhich is a block. This implies thatz is not visible from L giving a contradiction.
LettingTn be the number of squares inCn which meet VL(E) we have (3.2) dimB(VL(E)) = lim sup
n→∞
logTn
−log 2−n. Moreover, by (3.1), Tn ≤ Sn where Sn = P2n−1
j=0 Yn(j). Hence, it remains to estimate Sn.
By independence, the expectation of Yn(j) conditioned on Xn(j) =t is given by
E(Yn(j)|Xn(j) =t) =
t
X
i=1
iq(1−q)i−1+t(1−q)t<
∞
X
i=1
iq(1−q)i−1 =q−1. Since this upper bound is independent of t, we deduce E(Yn(j))< q−1. By linearity of the expectation, we have E(Sn)< q−12n.Lettingδ >0, Markov’s inequality gives
P(Sn>2n(1+δ))< q−12−nδ.
It follows from the Borel-Cantelli lemma that almost surely Sn ≤2n(1+δ) for all but finitely many n. Combining this with (3.2) shows that
dimB(VL(E))≤1 +δ.
Letting δ→0 implies that dimB(VL(E))≤1 almost surely.
For the Hausdorff measure assertion, note that H1(VL(E))≤lim inf
n→∞
√
22−nTn. By Fatou’s lemma,
E(H1(VL(E)))≤lim inf
n→∞
√
22−nE(Tn)≤√ 2q−1 giving H1(VL(E))<∞ almost surely, as desired.
When determining the almost sure value of dimensions of visible parts in the general case, the proof is more complicated than in the above case.
Note that the first claim of Lemma 2.1 is not valid for lines L which are not parallel to thex-axis. In fact,PL(E) =PL(Q0) with probability zero. However, in [1] we show that if the projection of E contains an interval with positive
probability, then it contains an interval of length arbitrarily close to that of the projection of Q0 with positive probability. Using this result we define blocks and windows similarly to the case of the x-axis. However, in the general case one can see through a block near the corners. Roughly speaking, the technical difficulties are due to the fact that being a corner is a deterministic property whilst being a block or a window is a random one. For details see [1].
Acknowledgements.We acknowledge the support of the Academy of Finland.
REFERENCES
[1] I. Arhosalo, E. J¨arvenp¨a¨a, M. J¨arvenp¨a¨a and P. Shmerkin,Visible parts of fractal per- colation. In preparation.
[2] M. Breen,Improved Krasnoselskii theorems for the dimension of the kernel of a star- shaped set. J. Geom.27(1986), 174–179.
[3] J. Cel,An optimal Krasnoselskii-type theorem for an open starshaped set. Geom. Dedi- cata66(1997), 293–301.
[4] L. Chayes, Aspects of the fractal percolation process. In: C. Bandt, S. Graf and M.
Z¨ahle (Eds.), Fractal Geometry and Stochastics (Finsterbergen, 1994), pp. 113–143.
Birkh¨auser, Basel, 1995.
[5] J.T. Chayes, L. Chayes and R. Durrett,Connectivity properties of Mandelbrot’s perco- lation process. Probab. Theory Related Fields77(1988), 307–324.
[6] M. Cs¨ornyei,On the visibility of invisible sets. Ann. Acad. Sci. Fenn. Math.25(2000), 417–421.
[7] R. Davies and H. Fast,Lebesgue density influences Hausdorff measure; large sets surface- like from many directions. Mathematika25(1978), 116–119.
[8] K.J. Falconer, The dimension of the convex kernel of a compact starshaped set. Bull.
London Math. Soc.9(1977), 313–316.
[9] K.J. Falconer and G.R. Grimmet,On the geometry of random Cantor sets and fractal percolation. J. Theoret. et Probab.5(1992), 465–485.
[10] K.J. Falconer and G.R. Grimmet, On the geometry of random Cantor sets and fractal percolation. J. Theoret. et Probab.7(1994), 209–210.
[11] G. Grimmett, “Percolation”, 2nd Edition. Springer-Verlag, Berlin, 1999.
[12] E. J¨arvenp¨a¨a, M. J¨arvenp¨a¨a, P. MacManus and T.C. O’Neil,Visible parts and dimen- sions. Nonlinearity16(2003), 803–818.
[13] E. J¨arvenp¨a¨a, M. J¨arvenp¨a¨a and J. Niemel¨a,Transversal mappings between manifolds and non-trivial measures on visible parts. Real Anal. Exchange25(2000), 629–640.
[14] R. Kaufman,On Hausdorff dimension of projections. Mathematika15(1968), 153–155.
[15] J.M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3)4(1954),3, 257–302.
[16] P. Mattila, Hausdorff dimension, projections, and the Fourier transform. Publ. Math.
48(2004), 3–48.
[17] P. Mattila,Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifia- bility. Cambridge Studies in Advanced Mathematics44. Cambridge Univ. Press, 1995.
[18] P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes.
Ann. Acad. Sci. Fenn. Ser. A I Math.1(1975), 227–244.
[19] T.C. O’Neil, The Hausdorff dimension of visible sets of planar continua. Trans. Amer.
Math. Soc.359(2007), 5141–5170.
[20] Y. Peres and W. Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J.102(2000), 193–251.
Received 8 December 2008 University of Oulu
Department of Mathematical Sciences P.O. Box 3000
FI-90014 Oulu, Finland [email protected]
