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On fair entropy of the tent family

  • * Corresponding author: Rui Gao

    * Corresponding author: Rui Gao

BG was partially supported by the Fundamental Research Funds for the Central Universities in China, and by National Natural Science Foundation of China (No. 12071118). RG was partially supported by the National Natural Science Foundation of China (No. 11701394)

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  • The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues [13] recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy $ h(a) $ of the tent map $ f_a $, as a function of the parameter $ a = \exp(h_{top}(f_a)) $, is continuous and strictly increasing on $ [\sqrt{2},2] $. In this short note, we extend the last result and characterize regularity of the function $ h $ precisely. We prove that $ h $ is $ \frac{1}{2} $-Hölder continuous on $ [\sqrt{2},2] $ and identify its best Hölder exponent on each subinterval of $ [\sqrt{2},2] $. On the other hand, parallel to a recent result on topological entropy of the quadratic family due to Dobbs and Mihalache [7], we give a formula of pointwise Hölder exponents of $ h $ at parameters chosen in an explicitly constructed set of full measure. This formula particularly implies that the derivative of $ h $ vanishes almost everywhere.

    Mathematics Subject Classification: Primary:37E05, 37A10.

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