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Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Applications of monotone spaces theory:Peano curves, Urbanski conjecture, differentiability. . .
Ondrej Zindulka
Czech Technical University Prague
Stara Lesna 2012
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Mapping a set of reals onto an interval
Theorem
Suppose C ⊆ R is compact, L(C) > 0.Then there is an onto Lipschitz mapping g : C → [0, 1].
Proof.
g(x) = L(C ∩ (−∞, x]
)g takes all values between 0 and L(C), hence g[C] contains an interval
|g(y)− g(x)| = L(C ∩ [x, y]
)6 |y − x|, i.e. g is Lipschitz
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Mapping a set of reals onto an interval
Theorem
Suppose C ⊆ R is compact, Hs(C) > 0.Then there is an onto s-Holder mapping g : C → [0, 1].
Proof.
Frostman Lemma: there is a Borel measure µ on C such that µ[x, y] 6 |x− y|s.
g(x) = µ(C ∩ (−∞, x])
g[C] contains an interval
|g(y)− g(x)| 6 µ[x, y] 6 |y − x|s, i.e. g is s-Holder
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Monotone metric spaces
Definition (Oz 2012)
A metric space (X, d) is
1-monotone if there is a linear order on X such that
x < y < z =⇒ d(x, y) 6 d(x, z),
monotone if it is Lipschitz equivalent to a 1-monotone space
Theorem (Keleti, Mathe, Oz 2012)
Suppose X is analytic and monotone. If Hs(X) > 0, then there is an onto s-Holder mappingg : C → [0, 1].
Corollary (Keleti, Mathe, Oz 2012)
Suppose X is analytic and monotone.If Hn(X) > 0, then there is an onto Lipschitz mapping g : X → [0, 1]n.
Theorem (Nekvinda, Oz 2011)
Every separable monotone space embeds into the line.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Lipschitz-ultrametric sets
Definition
A metric space (X, d) is
ultrametric if d(x, z) 6 max{d(x, y), d(y, z)
}.
Lipschitz-ultrametric if it is Lipschitz equivalent to an ultrametric space.
Theorem (Mendel, Naor 2012)
Let X be an analytic metric space. For every ε > 0 there is a compact Lipschitz-ultrametric setZ ⊆ X such that
dimH Y > dimH X − ε.
Theorem (Nekvinda, Oz 2011)
Every Lipschitz-ultrametric space is monotone.
Corollary (Keleti, Mathe, Oz 2012)
Suppose X is analytic. If dimH X > n, then there is an onto Lipschitz mapping g : X → [0, 1]n.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Transitive Hausdorff dimension
Definition (Transitive Hausdorff dimension — Urbanski 2009)
Let X be a separable metric space. Define
tHDX = sup{dim f [X] : f Lipschitz}
Proposition (Hurewicz–Wallman 1941)
tHDX 6 dimH X
Urbanski Conjecture
If X is a metric space with finite Hausdorff dimension, then
tHDX = bdimH Xc or tHDX = bdimH Xc − 1.
Failure
Urbanski Conjecture consistently fails: If ∃A ⊆ R2 such that |A| < c and L2(A) > 0.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Urbanski Conjecture
Corollary (Keleti–Mathe–Oz 2012)
Suppose X is analytic.If dimH X > n, then there is an onto Lipschitz mapping g : X → [0, 1]n.
Theorem (Keleti–Mathe–Oz 2012)
Suppose X is analytic.
If dimH X is finite but not an integer, then tHDX = bdimH Xc,if dimH X is an integer, then tHDX = dimH X or tHDX = dimH X − 1,
if dimH X =∞, then tHDX =∞.
Therefore Urbanski Conjecture holds for analytic spaces.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Failure of Urbanski Conjecture?
Theorem (Keleti–Mathe–Oz 2012)
There exist separable metric spaces with zero transfinite Hausdorff dimension and arbitrarily largeHausdorff dimension.
Question
Is Urbanski Conjecture consistently true?
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Transfinite Hausdorff dimension revisited
Proposition
tHDX > 1 if and only if there is an onto Lipschitz mapping f : X → [0, 1].
Proof.
Enough to show: If dimX > 0, then there is an onto Lipschitz mapping f : X → [0, 1].
If dimX > 0, there is x0 ∈ X such that all small circles centered at x0 are nonempty.
Define f(x) = d(x0, x).
Question
Is it true that tHDX > n if and only if there is an onto Lipschitz mapping f : X → [0, 1]n?
Theorem (Oz)
Yes if X is σ-compact.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Laczkovich Conjecture
Laczkovich Conjecture
Let X ⊆ Rn be compact. If Ln(X) > 0, then X maps by a Lipschitz map onto a ball.
Status of Laczkovich Conjecture
trivially holds for n = 1
[Preiss ∼1995] very nontrivially holds for n = 2
[Csornyei, Jones] hopefully holds for n > 3
Theorem (Oz)
Suppose Laczkovich Conjecture holds. If X is analytic, thentHDX > n if and only if there is an onto Lipschitz mapping f : X → [0, 1]n.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Peano curves
Peano curves
Peano curve is an onto continuous mapping p : [0, 1]m → [0, 1]n.
Holder mapping
A mapping is β-Holder if d(fx, fy) 6 Cd(x, y)β for some constant C
Holder Peano curves
There exist 12
-Holder Peano curves [0, 1]→ [0, 1]2
There exist 1n
-Holder Peano curves [0, 1]→ [0, 1]n
Question
Is there a 23
-Holder Peano curve [0, 1]2 → [0, 1]3?
Find S23 = sup{β : there is a β-Holder Peano curve [0, 1]2 → [0, 1]3}
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Peano curves: partial answer
Nearly Holder
A mapping is
nearly s-Holder if it is β-Holder for all β < s,
nearly Lipschitz if it is nearly 1-Holder.
Theorem
Let X ⊆ Rn be analytic, S ⊆ Rm self-similar and s = dimH S.If Hs(X) > 0, then there is a nearly Lipschitz mapping f : Rn → Rm such that S ⊆ f(X).
Corollary
Let X ⊆ Rm be analytic.If Hs(X) > 0, then there is an onto, nearly s
n-Holder mapping f : X → [0, 1]n.
Corollary
There is a nearly mn
-Holder Peano curve [0, 1]m → [0, 1]n.
In particular, S23 =2
3.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Universal measure zero sets with large Hausdorff dimension
Universal measure zero
A separable metric space X has universal measure zero (UMZ) if there is no diffused Borel measureµ, 0 < µ(X) <∞.
Theorem (Oz 2005)
There is a UMZ set X ⊆ [0, 1] such that dimH X = 1.
Theorem (Oz)
Every analytic metric space contains a UMZ set of the same Hausdorff dimension.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Industry of monotone spaces: papers and topics
Papers
4 published
1 accepted
2 submitted
4 in preparation
1 PhD thesis
Topics
Urbanski Conjecture
Universal measure zero
Peano curves
cardinal invariants of porous sets
fractal properties
geometry of curves
differentiability of functions
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Industry of monotone spaces: people involved
People involved
Pieter Allaart
Michael Hrusak
Tamas Keleti
Andras Mathe
Tamas Matrai
Ales Nekvinda
Dusan Pokorny
Arturo Rodrigues
Vasek Vlasak
Oz
Nations involved
Americans, Czechs, Hungarians, Mexicans. But no Polish and no Slovaks!
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Monotone sets in the plane
c-monotone sets
X ⊆ R2 is c-monotone if there is a linear order on X such that
x < y < z =⇒ |y − x| 6 c · |z − x|.
X is monotone ⇐⇒ X is c-monotone for some c.
Theorem (Hrusak, Oz 2011)
If X ⊆ R2 is monotone, then it is porous and dimH X < 2.
Example (Nekvinda, Oz 2011)
There is a set Cantor set C ⊆ R2 with the following properties:
H1(C) = 1,
if Y ⊆ C is monotone, then H1(Y ) = 0 and Y is nowhere dense in C,
there is Y ⊆ C closed such that dimH Y = 0 and Y is not a countable union of monotone sets.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Non-σ-monotone dust in the plane
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Curves
Question
Do monotone curves have low Hausdorff dimension?
Proposition (Pieter Allaart, Oz)
Koch curve is monotone. The optimal witnessing constant is c =
√490747
277636.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Fuctions with monotone graphs
Framework
f : I → R is a continuous function on an interval I
Gr(f) is the graph of f
ψ(x) = (x, f(x)) is the natural parametrization of the graph
Dini derivatives
Right upper Dini derivative: f+(x)
Derivative: f ′(x)
D-point: f ′(x) exists
D is the set of all D-points
knot point: left and right derivatives do not exist, all Dini derivatives infinite
K is the set of all knot points
Approximate Dini derivatives
f+app(x), f
′app(x), Dapp etc.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
M-points
Vague question
How is differentiability of f related to monotonicity of Gr(f)?
Definition (M-points)
y ∈ I is an M-point if there are c and ε > 0 such that
x ∈ (y − ε, y), z ∈ (y, y + ε) =⇒ |ψ(y)− ψ(x)| 6 c|ψ(z)− ψ(x)|.
M is the set of all M-points.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
M-points
Theorem (Oz, Hrusak, Matrai, Nekvinda, Vlasak 201?)
There is a set E ⊆ I such that H1(Gr(f |E)) = 0 and
D ⊆M ⊆ Dapp ∪Kapp ∪E.
Therefore
Every D-point is an M-point.
Almost every M-point is either an approximate D-point or a knot point.
Theorem (Oz, Hrusak, Matrai, Nekvinda, Vlasak 201?)
The set Gr(f |M) has σ-finite linear measure. In particular, dimH Gr(f |M) = 1.
Corollary
If Gr(f) is monotone, then Gr(f) has σ-finite linear measure and thus dimH Gr(f) = 1.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Differentiability when the graph is monotone
Proposition
If Gr(f) is monotone, then f+app(a) = f+(a) for all a ∈ [0, 1]. Likewise for all Dini derivatives.
Theorem
If Gr(f) is monotone, then almost every point is either a D-point or a knot point.
Theorem
If Gr(f) is monotone, then every interval contains a perfect set of D-points.In particular, f is differentiable at a dense set.
D-points vs. M-points
D ⊆M ⊆ess Dapp ∪KintM⊆ D
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Some examples
Let ‖x‖ = dist(x,Z) be the triangle wave function.
Example
Let f(x) =∑∞k=0 2
−k‖2k2
x‖.f is continuous,
f has no M-points,
every monotone subset of Gr(f) is meager,
f is nowhere differentiable.
Example
Let g(x) = (x− 12) sin 1
2x−1f(x).
g is continuous,
g has exactly one M-point,
g is nowhere differentiable,
D is not dense in M.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Some examples
Example
Let f(x) =∑∞n=0 2
−n‖2nx‖ be the Takagi function.
f is continuous,
f does not possess a finite one-sided derivative at any point,
if x is a dyadic rational, then f+(x) = +∞ and f−(x) = −∞,
f ′(x) = +∞ at a dense set,
D is dense and co-dense,
M is dense and co-dense.
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
Nondifferentiable function with monotone graph
Theorem
There is a continuous, almost nowhere differentiable function f : [0, 1]→ R with a monotone graph.
Properties of the function
Every point is an M-point,
the function is almost nowhere approximately differentiable,
almost all points are knot points (actually approximate knot points),
the function has a derivative at a perfectly dense set.
Theorem
There is an absolutely continuous function f such that M is co-dense.
Properties of the function
f is differentiable almost everywhere,
almost all points are M-points,
all monotone subsets of Gr(f) are nowhere dense (in Gr(f)).
Ondrej Zindulka Applications of monotone spaces theory
Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture
A conjecture
Theorem
If f : [0, 1]→ R has a 1-monotone graph, then it is of bounded variation.
Theorem (Nekvinda, Pokorny, Vlasak)
If X ⊆ Rn is 1-monotone, then dimH X = 1.
Conjecture 1
There is a function f : [1,∞)→ [1, 2] such that
limc→1+ f(c) = 1,
if X ⊆ R2 is c-monotone, then dimH X 6 f(c).
Conjecture 2
The same, but for c-monotone curves in R2.
Ondrej Zindulka Applications of monotone spaces theory