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August  2021, 41(8): 3797-3816. doi: 10.3934/dcds.2021017

On fair entropy of the tent family

Bing Gao 1, and  Rui Gao 2,,

1. 

School of Mathematics, Hunan University, Changsha 410082, China
2. 

College of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author: Rui Gao

Received  August 2020 Revised  December 2020 Published  August 2021 Early access  January 2021

Fund Project: 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)

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.

Keywords: Tent map, fair measure, fair entropy, Hölder exponent, equilibrium state.
Mathematics Subject Classification: Primary:37E05, 37A10.
Citation: Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3797-3816. doi: 10.3934/dcds.2021017
References:
[1]

V. Baladi, Positive Transfer Operators and Decay of Correlations, volume 16 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

[2]

V. Baladi and D. Ruelle, An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps, Ergodic Theory Dynam. Systems, 14 (1994), 621-632.  doi: 10.1017/S0143385700008087.  Google Scholar

[3]

O. F. Bandtlow and H. H. Rugh, Entropy continuity for interval maps with holes, Ergodic Theory Dynam. Systems, 38 (2018), 2036-2061.  doi: 10.1017/etds.2016.115.  Google Scholar

[4]

K. Brucks and M. Misiurewicz, The trajectory of the turning point is dense for almost all tent maps, Ergodic Theory Dynam. Systems, 16 (1996), 1173-1183.  doi: 10.1017/S0143385700009962.  Google Scholar

[5]

H. Bruin, For almost every tent map, the turning point is typical, Fund. Math., 155 (1998), 215-235.   Google Scholar

[6]

E. M. Coven, I. Kan and J. A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc., 308 (1988), 227-241.  doi: 10.1090/S0002-9947-1988-0946440-2.  Google Scholar

[7]

N. Dobbs and N. Mihalache, Diabolical entropy, Comm. Math. Phys., 365 (2019), 1091-1123.  doi: 10.1007/s00220-019-03293-y.  Google Scholar

[8]

M. Keane, Strongly mixing $g$-measures, Invent. Math., 16 (1972), 309-324.  doi: 10.1007/BF01425715.  Google Scholar

[9]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152, http://www.numdam.org/item/?id=ASNSP_1999_4_28_1_141_0.  Google Scholar

[10]

F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 185-202.  doi: 10.1007/BF00533471.  Google Scholar

[11]

C. Liverani, B. Saussol and S. Vaienti, Conformal measure and decay of correlation for covering weighted systems, Ergodic Theory Dynam. Systems, 18 (1998), 1399-1420.  doi: 10.1017/S0143385798118023.  Google Scholar

[12]

J. Milnor and W. Thurston, On iterated maps of the interval, In Dynamical Systems (College Park, MD, 1986-87), volume 1342 of Lecture Notes in Math., Springer, Berlin, 1988, 465-563. doi: 10.1007/BFb0082847.  Google Scholar

[13]

M. Misiurewicz and A. Rodrigues, Counting preimages, Ergodic Theory Dynam. Systems, 38 (2018), 1837-1856.  doi: 10.1017/etds.2016.103.  Google Scholar

[14]

H. H. Rugh and L. Tan, Kneading with weights, J. Fractal Geom., 2 (2015), 339-375.  doi: 10.4171/JFG/24.  Google Scholar

[15]

G. Tiozzo, Continuity of core entropy of quadratic polynomials, Invent. Math., 203 (2016), 891-921.  doi: 10.1007/s00222-015-0605-9.  Google Scholar

[16]

G. Tiozzo, The local Hölder exponent for the entropy of real unimodal maps, Sci. China Math., 61 (2018), 2299-2310.  doi: 10.1007/s11425-017-9293-7.  Google Scholar

[17]

P. Walters, Ruelle's operator theorem and $g$-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387.  doi: 10.2307/1997113.  Google Scholar

show all references

References:
[1]

V. Baladi, Positive Transfer Operators and Decay of Correlations, volume 16 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

[2]

V. Baladi and D. Ruelle, An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps, Ergodic Theory Dynam. Systems, 14 (1994), 621-632.  doi: 10.1017/S0143385700008087.  Google Scholar

[3]

O. F. Bandtlow and H. H. Rugh, Entropy continuity for interval maps with holes, Ergodic Theory Dynam. Systems, 38 (2018), 2036-2061.  doi: 10.1017/etds.2016.115.  Google Scholar

[4]

K. Brucks and M. Misiurewicz, The trajectory of the turning point is dense for almost all tent maps, Ergodic Theory Dynam. Systems, 16 (1996), 1173-1183.  doi: 10.1017/S0143385700009962.  Google Scholar

[5]

H. Bruin, For almost every tent map, the turning point is typical, Fund. Math., 155 (1998), 215-235.   Google Scholar

[6]

E. M. Coven, I. Kan and J. A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc., 308 (1988), 227-241.  doi: 10.1090/S0002-9947-1988-0946440-2.  Google Scholar

[7]

N. Dobbs and N. Mihalache, Diabolical entropy, Comm. Math. Phys., 365 (2019), 1091-1123.  doi: 10.1007/s00220-019-03293-y.  Google Scholar

[8]

M. Keane, Strongly mixing $g$-measures, Invent. Math., 16 (1972), 309-324.  doi: 10.1007/BF01425715.  Google Scholar

[9]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152, http://www.numdam.org/item/?id=ASNSP_1999_4_28_1_141_0.  Google Scholar

[10]

F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 185-202.  doi: 10.1007/BF00533471.  Google Scholar

[11]

C. Liverani, B. Saussol and S. Vaienti, Conformal measure and decay of correlation for covering weighted systems, Ergodic Theory Dynam. Systems, 18 (1998), 1399-1420.  doi: 10.1017/S0143385798118023.  Google Scholar

[12]

J. Milnor and W. Thurston, On iterated maps of the interval, In Dynamical Systems (College Park, MD, 1986-87), volume 1342 of Lecture Notes in Math., Springer, Berlin, 1988, 465-563. doi: 10.1007/BFb0082847.  Google Scholar

[13]

M. Misiurewicz and A. Rodrigues, Counting preimages, Ergodic Theory Dynam. Systems, 38 (2018), 1837-1856.  doi: 10.1017/etds.2016.103.  Google Scholar

[14]

H. H. Rugh and L. Tan, Kneading with weights, J. Fractal Geom., 2 (2015), 339-375.  doi: 10.4171/JFG/24.  Google Scholar

[15]

G. Tiozzo, Continuity of core entropy of quadratic polynomials, Invent. Math., 203 (2016), 891-921.  doi: 10.1007/s00222-015-0605-9.  Google Scholar

[16]

G. Tiozzo, The local Hölder exponent for the entropy of real unimodal maps, Sci. China Math., 61 (2018), 2299-2310.  doi: 10.1007/s11425-017-9293-7.  Google Scholar

[17]

P. Walters, Ruelle's operator theorem and $g$-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387.  doi: 10.2307/1997113.  Google Scholar

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