Hausdorff Dimension of Measures¶via Poincaré Recurrence

2001 ◽  
Vol 219 (2) ◽  
pp. 443-463 ◽  
Author(s):  
L. Barreira ◽  
B. Saussol
Keyword(s):  
Hausdorff Dimension ◽  
Poincare Recurrence ◽  
Poincaré Recurrence

SHRINKING TARGET PROBLEM FOR RANDOM IFS

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850085
Author(s):  
ZHIHUI YUAN
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Main Idea ◽  
Iterated Function Systems ◽  
Target Problem ◽  
Iterated Function ◽  
Poincare Recurrence ◽  
Poincaré Recurrence

We describe the shrinking target problem for random iterated function systems which are semi-conjugate to random subshifts. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The main idea is an extension of ubiquity theorem which plays an important role to get the lower bound of the dimension. Our method can be used to deal with the sets with respect to more general targets and the sets based on the quantitative Poincaré recurrence properties.

SPECTRA OF RECURRENCE DIMENSION FOR ADIC SYSTEMS

2002 ◽  
Vol 02 (04) ◽  
pp. 599-607 ◽  
Author(s):  
VÍCTOR F. SIRVENT
Keyword(s):  
Finite Type ◽  
Subshift Of Finite Type ◽  
Poincare Recurrence ◽  
Definition Of ◽  
Poincaré Recurrence

We compute the spectra of the recurrence dimension for adic systems and sub-adic systems. This dimension is characterized by the Poincaré recurrence of the system, and known in the literature as Afraimovich–Pesin dimension. These spectra are invariant under bi-Lipschitz transformations. We show that there is a duality between the spectra of an adic system and the corresponding subshift of finite type. We also consider Billingsley-like definition of the spectra of the recurrence dimension.

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