Topological pressure and fractal dimensions of cookie-cutter-like sets

Details Topological pressure and fractal dimensions of cookie-cutter-like sets Topological pressure and fractal dimensions of cookie-cutter-like sets

2012-03-13

arxiv.org/abs/1203.2724v3

Mathematics

Dynamical Systems

Scientific paper

The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter-like set first we have determined the topological pressure function, and then by Banach limit we have determined a unique Borel probability measure $\mu_h$ with the support the cookie-cutter-like set $E$. With the topological pressure and the measure $\mu_h$, we have shown that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero $h$ of the pressure function. Moreover, we have shown that the $h$-dimensional Hausdorff measure and the $h$-dimensional packing measure are finite and positive.

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