The shrinking target problem for self-affine systems

  • 2 November 2022
  • 1400-1500
  • Sch 0.13

Thomas Jordan (Bristol)

The shrinking target problem looks at a dynamical system \(T:X\to X\), a sequence of shrinking (in some suitable sense) sets \(\{B_n\}\) and the set of points \(x\) for which \(T^n(x)\in B_n\) for 

infinitely many \(n\in\N\). A typical problem is to look at whether the sets have 
full measure for some natural invariant measure and in the case where the set 
has zero measure to look at the Hausdorff dimension. We look at the second 
problem for a self-affine dynamical system in \(\R^2\). We take the set of \(B_n\) 
to be geometric balls. It turns out that how the dimension of the shrinking 
target set behaves depends on the centre of the ball. The techniques we need to 
use to the study these sets come from the study of Bernoulli convolutions and 
include both the transversality technique and the more recent results of 
Hochman and Shmerkin. This is joint work with Henna Koivusalo.

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