I will present our recent result, joint with Henna Koivusalo and Lingmin Liao, on the shrinking target set (for a generic affine iterated function system) in which the size of the targets is not fixed but depends on the trajectory. The simplest way of describing this set is to look at its presentation in the symbolic space: it is (corresponds to) the set of infinite symbolic sequences $i\in \Sigma = \{1,\ldots,N\}^{\mathbb N}$ such that for infinitely many $n\in \mathbb N$ we have \[i_{n+1} \ldots i_{n+m} = j_1 \ldots j_m\] for a fixed $j\in \Sigma$ and for $m=m(i,n)$ satisfying certain almost-additivity condition. For example, we can take $m$ as the partial Birkhoff sum of some continuous potential: $m(i,n) = \sum_{k=0}^n \xi(\sigma^k i)$.

The proof uses noncommutative thermodynamical formalism, in particular I will present a method (coming from Bárány and Troscheit) of constructing of the thermodynamical formalism for weakly quasi-additive potentials.

Meeting ID: 842 4054 6345 Passcode: 023053