Intermediate Dimensions, Capacities and Projections Kenneth - PowerPoint PPT Presentation

Intermediate Dimensions, Capacities and Projections Kenneth Falconer University of St Andrews, Scotland, UK Joint with Stuart Burrell, Jon Fraser and Tom Kempton Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Overview • The talk concerns sets in R n with differing Hausdorff and box-counting dimensions. • Hausdorff and box-counting dimensions can be regarded as particular cases of a spectrum of ‘intermediate’ dimensions dim θ F (0 ≤ θ ≤ 1) with dim 0 F = dim H F and dim 1 F = dim B F • Intermediate dimensions give an idea of the range of sizes of covering sets needed to get good estimates for Hausdorff dimension. • Potential theoretic methods enable us to study geometric properties of these dimensions such as the effect of orthogonal projection. Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Hausdorff and box dimension - alternative definitions Recall that Hausdorff dimension may be defined without introducing Hausdorff measures: for E ⊂ R n � dim H E = inf s ≥ 0 : for all ǫ > 0 there exists a cover { U i } of E such that � | U i | s ≤ ǫ � . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Hausdorff and box dimension - alternative definitions Recall that Hausdorff dimension may be defined without introducing Hausdorff measures: for E ⊂ R n � dim H E = inf s ≥ 0 : for all ǫ > 0 there exists a cover { U i } of E such that � | U i | s ≤ ǫ � . The lower/upper box-counting dimensions of a non-empty compact E ⊂ R n are log N r ( E ) log N r ( E ) dim B E = lim inf , dim B E = lim − log r − log r r → 0 r → 0 where N r ( E ) is the least number of sets of diameter r covering E . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Hausdorff and box dimension - alternative definitions Recall that Hausdorff dimension may be defined without introducing Hausdorff measures: for E ⊂ R n � dim H E = inf s ≥ 0 : for all ǫ > 0 there exists a cover { U i } of E such that � | U i | s ≤ ǫ � . The lower/upper box-counting dimensions of a non-empty compact E ⊂ R n are log N r ( E ) log N r ( E ) dim B E = lim inf , dim B E = lim − log r − log r r → 0 r → 0 where N r ( E ) is the least number of sets of diameter r covering E . Equivalently dim B may be defined � dim B E = inf s ≥ 0 : for all ǫ > 0 there exists a cover { U i } of E such that | U i | = | U j | for all i , j and � | U i | s ≤ ǫ � . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Intermediate dimensions Let E ⊂ R n be non-empty and bounded. For 0 ≤ θ ≤ 1 define the lower θ -intermediate dimension of E by � dim θ E = inf s ≥ 0 : for all ǫ > 0 there exist arbitrarily small δ > 0 s.t. and { U i } covering E s.t. δ 1 /θ ≤ | U i | ≤ δ and � | U i | s ≤ ǫ � . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Intermediate dimensions Let E ⊂ R n be non-empty and bounded. For 0 ≤ θ ≤ 1 define the lower θ -intermediate dimension of E by � dim θ E = inf s ≥ 0 : for all ǫ > 0 there exist arbitrarily small δ > 0 s.t. and { U i } covering E s.t. δ 1 /θ ≤ | U i | ≤ δ and � | U i | s ≤ ǫ � . Similarly, define the upper θ -intermediate dimension of E by � dim θ E = inf s ≥ 0 : for all ǫ > 0 and all sufficiently small δ > 0 there is a cover { U i } of E s.t. δ 1 /θ ≤ | U i | ≤ δ and � | U i | s ≤ ǫ � . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Intermediate dimensions Let E ⊂ R n be non-empty and bounded. For 0 ≤ θ ≤ 1 define the lower θ -intermediate dimension of E by � dim θ E = inf s ≥ 0 : for all ǫ > 0 there exist arbitrarily small δ > 0 s.t. and { U i } covering E s.t. δ 1 /θ ≤ | U i | ≤ δ and � | U i | s ≤ ǫ � . Similarly, define the upper θ -intermediate dimension of E by � dim θ E = inf s ≥ 0 : for all ǫ > 0 and all sufficiently small δ > 0 there is a cover { U i } of E s.t. δ 1 /θ ≤ | U i | ≤ δ and � | U i | s ≤ ǫ � . Then dim 0 E = dim 0 E = dim H E , dim 1 E = dim B E and dim 1 E = dim B E . Moreover, for bounded E and θ ∈ [0 , 1], dim H E ≤ dim θ E ≤ dim θ E ≤ dim B E and dim θ E ≤ dim B E . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

SImple properties • dim θ is finitely stable, that is dim θ ( E 1 ∪ E 2 ) = max { dim θ E 1 , dim θ E 2 } . • For θ ∈ (0 , 1], both dim θ E and dim θ E are unchanged on replacing E by its closure. • For E , F ⊆ R n be non-empty and bounded and θ ∈ [0 , 1], dim θ E +dim θ F ≤ dim θ ( E × F ) ≤ dim θ ( E × F ) ≤ dim θ E +dim B F . • For θ ∈ [0 , 1], dim θ and dim θ are bi-Lipschitz invariant. Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Continuity and monotonicity Proposition Let E ⊂ R n and let 0 ≤ θ < φ ≤ 1. Then 1 − θ � � dim θ E ≤ dim φ E ≤ dim θ E + ( n − dim θ E ) , φ similarly for upper dimensions. In particular, θ �→ dim θ E and θ �→ dim θ E are continuous for θ ∈ (0 , 1] and (not necessarily strictly) increasing. Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Intermediate dimensions and Assouad dimension The Assouad dimension of E ⊆ R n is defined by � dim A E = inf s ≥ 0 : there exists C > 0 such that for all x ∈ E , � s � � R and for all 0 < r < R , N r ( E ∩ B ( x , R )) ≤ C r where N r ( A ) denotes the smallest number of sets of diameter at most r required to cover a set A . In general dim B E ≤ dim B E ≤ dim A E ≤ n , Proposition For non-empty bounded E ⊆ R n and θ ∈ (0 , 1], dim θ E ≥ dim A E − dim A E − dim B E , θ with a similar conclusion using dim θ and dim B . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Example For p > 0 let 0 , 1 1 p , 1 2 p , 1 � � E p = 3 p , . . . . E 1 : Since E p is countable, dim H E p = 0. It is well-known that dim B E p = 1 / ( p + 1). For p > 0 and 0 ≤ θ ≤ 1, θ dim θ E p = dim θ E p = p + θ. Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Examples � � E log = 0 , 1 / log 2 , 1 / log 3 , . . . E 1 ∪ E where dim H E = dim B E = 1 / 3 Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Examples E 1 ∪ E where E 1 × E log dim B E = dim A E = 1 / 4 Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Bedford-McMullen carpets 3 × 4 Bedford-McMullen self-affine carpet Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Bedford-McMullen carpets 2 × 3 and 3 × 5 Bedford-McMullen self-affine carpets Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Bedford-McMullen carpets p × q carpet, p < q (Bedford 1984, McMullen 1984) p 1 � � N log p / log q � dim H E = log p log j j =1 � p log 1 j =1 N j dim B E = log N N log p + log q N j rectangles selected in j th column, N non-empty columns. Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Bedford-McMullen carpets Proposition Let E be the Bedford-McMullen carpet as above. Then for 0 < θ < 1 4 (log p / log q ) 2 , � 2 log(log p / log q ) log(max j N j ) � 1 dim θ E ≤ dim H E + − log θ. log q (1) In particular, dim θ E and dim θ E are continuous at θ = 0 and so are continuous on [0 , 1]. Proof Put a natural Bernoulli measure µ on E and show that for all x ∈ E , µ ( S ( x , p − k )) ≥ ( p − k ) d + ǫ for some K ≤ k ≤ K /θ for all large K , where S ( x , p − k ) is an ‘approximate square’ of centre x and side p − k . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Bedford-McMullen carpets Proposition Let E be the Bedford-McMullen carpet as above. Then for 0 ≤ θ ≤ log p / log q , log � p j =1 N j − H ( µ ) dim θ E ≥ dim H E + θ . (2) log p where H ( µ ) < log � p j =1 N j is the entropy of the Bernoulli measure on E . Proof For each K , construct a measure ν K on E and show that for ν K ( S ( x , p − k )) ≤ ( p − k ) d ′ − ǫ for some E 0 ⊂ E with ν K ( E 0 ) ≥ 1 2 , all x ∈ E 0 and K ≤ k ≤ K /θ . Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Bedford-McMullen carpets Lower bound for dim θ E , upper bound for dim θ E Kenneth Falconer Intermediate Dimensions, Capacities and Projections

Marstrand’s projection theorems Theorem (Marstrand 1954, Mattila 1975) Let E ⊂ R n be Borel. For all α ∈ G ( n , m ) dim H proj α E ≤ min { dim H E , m } ≡ dim m H E with equality for almost all α ∈ G ( n , m ), [proj α is orthogonal projection onto the m -dimensional subspace α ] Think of dim m H E as ‘the dimension of E when viewed from an m -dimensional viewpoint’ or the m -dimensional Hausdorff dimension profile of E . Kenneth Falconer Intermediate Dimensions, Capacities and Projections