How inhomogeneous Cantor sets can pass a point

Abstract: Abstract: For $x > 0$, we define $$\Upsilon(x) = \{ (a,b): x\in E_{a,b}, a>0, b>0, a+b \le 1 \},$$ where the set $E_{a,b}$ is the unique nonempty compact invariant set generated by the inhomogeneous IFS $$\{ f_0(x) = a x, f_1(x) = b(x+1) \}.$$ We show the set $\Upsilon(x)$ is a Lebesgue null set with full Hausdorff dimension in $\mathbb{R}^2$, and the intersection of sets $\Upsilon(x_1), \Upsilon(x_2), \dots, \Upsilon(x_\ell)$ still has full Hausdorff dimension $\mathbb{R}^2$ for any finitely many positive real numbers $x_1, x_2, \dots, x_\ell$.

dynamical systemsnumber theory

Audience: researchers in the topic

( slides )

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One World Numeration seminar

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