Projections of Mandelbrot percolations Michał Rams 1 Károly Simon 2 1 Institute of Mathematics Polish Academy of Sciences Warsaw, Poland http://www.impan.pl/~rams/ 2 Department of Stochastics Institute of Mathematics Technical University of Budapest www.math.bme.hu/~simonk 12 December 2012 Hong Kong Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 1 / 38
Outline History 1 The projections 2 Percolation phenomenon 3 New results 4 The sum of three linear random Cantor sets 5 Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 2 / 38
All new results are joint with Michal Rams, Warsaw IMPAN Michal visited me last week in Budapest and while we were preparing our joint talk, he got a terrible flu which prevented him from participating in this conference. Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 3 / 38
Outline History 1 The projections 2 Percolation phenomenon 3 New results 4 The sum of three linear random Cantor sets 5 Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 4 / 38
Fractal percolation, introduced by Mandelbrot early 1970’s: We partition the unit square into M 2 congruent sub squares each of them are independently retained with probability p and discarded with probability 1 − p . In the squares retained after the previous step we repeat the same process at infinitum. Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 5 / 38
Fractal percolation, introduced by Mandelbrot early 1970’s: We partition the unit square into M 2 congruent sub squares each of them are independently retained with probability p and discarded with probability 1 − p . In the squares retained after the previous step we repeat the same process at infinitum. Λ 1 Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 5 / 38
Fractal percolation, introduced by Mandelbrot early 1970’s: We partition the unit square into M 2 congruent sub squares each of them are independently retained with probability p and discarded with probability 1 − p . In the squares retained after the previous step we repeat the same process at infinitum. Λ 2 Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 5 / 38
Fractal percolation, introduced by Mandelbrot early 1970’s: We partition the unit square into M 2 congruent sub squares each of them are independently retained with probability p and discarded with probability 1 − p . In the squares retained after the previous step we repeat the same process at infinitum. Λ 3 Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 5 / 38
Let Λ n be the union of the level n retained squares. Then the statistically self-similar set of interest is: ∞ � Λ := Λ n . n = 1 It was proved by Falconer and independently Mauldin, Willims that conditioned on non-extinction: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M The expected number of descendants of every square is: M 2 · p . Therefore, if M 2 · p < 1 then Λ = ∅ a.s. Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 6 / 38
Let Λ n be the union of the level n retained squares. Then the statistically self-similar set of interest is: ∞ � Λ := Λ n . n = 1 It was proved by Falconer and independently Mauldin, Willims that conditioned on non-extinction: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M The expected number of descendants of every square is: M 2 · p . Therefore, if M 2 · p < 1 then Λ = ∅ a.s. Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 6 / 38
Let Λ n be the union of the level n retained squares. Then the statistically self-similar set of interest is: ∞ � Λ := Λ n . n = 1 It was proved by Falconer and independently Mauldin, Willims that conditioned on non-extinction: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M The expected number of descendants of every square is: M 2 · p . Therefore, if M 2 · p < 1 then Λ = ∅ a.s. Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 6 / 38
Let Λ n be the union of the level n retained squares. Then the statistically self-similar set of interest is: ∞ � Λ := Λ n . n = 1 It was proved by Falconer and independently Mauldin, Willims that conditioned on non-extinction: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M The expected number of descendants of every square is: M 2 · p . Therefore, if M 2 · p < 1 then Λ = ∅ a.s. Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 6 / 38
So, we have almost surely: If p ≤ 1 / M 2 then Λ = ∅ . If 1 / M 2 < p < 1 / M then dim H (Λ) < 1 (but Λ � = ∅ with positive probability). If p > 1 M then either (a) Λ = ∅ or (b) dim H (Λ) > 1 . Recall: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 7 / 38
So, we have almost surely: If p ≤ 1 / M 2 then Λ = ∅ . If 1 / M 2 < p < 1 / M then dim H (Λ) < 1 (but Λ � = ∅ with positive probability). If p > 1 M then either (a) Λ = ∅ or (b) dim H (Λ) > 1 . Recall: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 7 / 38
So, we have almost surely: If p ≤ 1 / M 2 then Λ = ∅ . If 1 / M 2 < p < 1 / M then dim H (Λ) < 1 (but Λ � = ∅ with positive probability). If p > 1 M then either (a) Λ = ∅ or (b) dim H (Λ) > 1 . Recall: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 7 / 38
So, we have almost surely: If p ≤ 1 / M 2 then Λ = ∅ . If 1 / M 2 < p < 1 / M then dim H (Λ) < 1 (but Λ � = ∅ with positive probability). If p > 1 M then either (a) Λ = ∅ or (b) dim H (Λ) > 1 . Recall: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 7 / 38
So, we have almost surely: If p ≤ 1 / M 2 then Λ = ∅ . If 1 / M 2 < p < 1 / M then dim H (Λ) < 1 (but Λ � = ∅ with positive probability). If p > 1 M then either (a) Λ = ∅ or (b) dim H (Λ) > 1 . Recall: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 7 / 38
So, we have almost surely: If p ≤ 1 / M 2 then Λ = ∅ . If 1 / M 2 < p < 1 / M then dim H (Λ) < 1 (but Λ � = ∅ with positive probability). If p > 1 M then either (a) Λ = ∅ or (b) dim H (Λ) > 1 . Recall: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 7 / 38
So, we have almost surely: If p ≤ 1 / M 2 then Λ = ∅ . If 1 / M 2 < p < 1 / M then dim H (Λ) < 1 (but Λ � = ∅ with positive probability). If p > 1 M then either (a) Λ = ∅ or (b) dim H (Λ) > 1 . Recall: dim H Λ = dim B Λ = log ( M 2 · p ) a.s. log M Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 7 / 38
Marstrand Theorem Theorem (Marstrand) Let B ⊂ R 2 be a Borel set. If dim H ( B ) ≤ 1 then for L eb -a.e. θ , we have 1 dim H ( proj θ ( B )) = dim H ( B ) If dim H ( B ) > 1 then for L eb -a.e. θ , we have 2 L eb ( proj θ ( B )) > 0 . Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 8 / 38
Marstrand Theorem Theorem (Marstrand) Let B ⊂ R 2 be a Borel set. If dim H ( B ) ≤ 1 then for L eb -a.e. θ , we have 1 dim H ( proj θ ( B )) = dim H ( B ) If dim H ( B ) > 1 then for L eb -a.e. θ , we have 2 L eb ( proj θ ( B )) > 0 . Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 8 / 38
Marstrand Theorem Theorem (Marstrand) Let B ⊂ R 2 be a Borel set. If dim H ( B ) ≤ 1 then for L eb -a.e. θ , we have 1 dim H ( proj θ ( B )) = dim H ( B ) If dim H ( B ) > 1 then for L eb -a.e. θ , we have 2 L eb ( proj θ ( B )) > 0 . Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 8 / 38
Outline History 1 The projections 2 Percolation phenomenon 3 New results 4 The sum of three linear random Cantor sets 5 Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 9 / 38
Orthogonal projection to ℓ θ Λ p r o j θ ( Λ ) θ ℓ θ Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 10 / 38
Radial and co-radial projections with center t C 1 Λ t Proj t (Λ) Let CProj t (Λ) := { dist ( t , x ) : x ∈ Λ } ( CProj t (Λ) is the set of the length of dashed lines above). Michał Rams, Károly Simon (IMPAN, TU Budapest) Projections of Mandelbrot percolations CUHK 11 / 38
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