2025/01/27 by Guiraud, P., Hernández, M., Meyroneinc, A. +1
#37B10 #37E05 #37E10 #37E45 #37G35 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.16263
We consider a family of piecewise contractions admitting a rotation number and defined for every x∈[0,1) by f(x)=λx + δ+ d θa(x) \pmod 1, where λ∈(0,1), d∈(0,1-λ), δ∈[0,1], a∈[0,1] and θa(x)=1 if x≥ a and θa(x)=0 otherwise. In the special case where a=1, the family reduces to the well studied ``contracted rotations" x↦ λx + δ\pmod 1, which are 2-interval piecewise λ-affine contractions when δ∈(1-λ,1). Considering a∈(0,1) allows maps with an additional discontinuity, that is, 3-interval piecewise λ-affine contractions. Supposing λ and d fixed, for any ρ∈(0,1) and α∈[0,1], we provide the values of the parameters δ and a for which the corresponding map has rotation number ρ, and a symbolic dynamics containing that of the rotation Rρ:[0,1)→[0,1) of angle ρ with respect to the partition given by the positions of 1-ρ and α in [0,1). This enables in particular to determine the maps that have a given number of periodic orbits of an arbitrary period, or a Cantor set attractor supporting a dynamics of a given complexity.