On minimal homeomorphisms on Peano continua

Following the question of Artigue we construct a minimal homeomorphism g: X→Xg: X→X on a Peano continuum X with the following property: there exist a positive number ε   and a dense GδGδ subset E of X such that every non-trivial subcontinuum of X intersecting E expands under iterations of g to a continuum of diameter greater than ε.