Topological fractals.

A compact metric space is said to be a topological fractal if there is a finite set of continuous selfmaps of that is topologically contractive and the images of these maps cover . In 1985 Hata observed that every connected topological fractal is a Peano continuum and conjectured that every Peano continuum is a topological fractal. To date, only partial solutions were found.
Following up on these we provide another partial solution together with Benjamin Vejnar, namely we prove that every Peano continuum with uncountably many local cut points is a topological fractal. We modify the set of selfmaps to reduce its size, reaching the optimal two-element witnessing set of selfmaps for Peano continua with uncountably many cut points and three-element witnessing set otherwise.