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