RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 123Figure 7: An example showing a case in which the stretching property is not satisfied.The white region is moved to the white one and the darker to the darker one by ahomeomorphism which does not have fixed points.In order to better understand which are the good crossings between R and ψ(R)that permit to apply our fixed point theorem, we considered the following definitionsof slabs of an oriented rectangle (called “ slices ” in [60]).Let ̂M = (M,M − ) and ̂N = (N,N − ) be two oriented cells in X. We say that̂M is a horizontal slab of ̂N and writêM ⊆ h ̂N,if M ⊆ N and, eitherorM − l⊆ N − land M − r ⊆ N − r ,M − l⊆ N − r and M − r ⊆ N − l.Similarly, we say that ̂M is a vertical slab of ̂N and writêM ⊆ v ̂N,if M ⊆ N and, eitherorM + b ⊆ N+ band M + t ⊆ N + t ,M + b ⊆ N+ t and M + t ⊆ N + b .