American Institute of Mathematical Sciences

Advanced
March  2021, 41(3): 1469-1482. doi: 10.3934/dcds.2020325

Global graph of metric entropy on expanding Blaschke products

Yunping Jiang ,

1. 

Department of Mathematics, Queens College of the City University of New York, Flushing, NY 11367-1597
2. 

Department of Mathematics, Graduate Center of the City University of New York, New York, NY 10016

* Corresponding author

Received  April 2019 Revised  July 2020 Published  March 2021 Early access  September 2020

Fund Project: This material is based upon work supported by the National Science Foundation. It is also partially supported by the Simons Foundation collaboration grant (grant number 523341) and the PSC-CUNY Enhanced Research Award (award number 62777-00 50)

We study the global picture of the metric entropy on the space of expanding Blaschke products. We first construct a smooth path in the space tending to a parabolic Blaschke product. We prove that the metric entropy on this path tends to 0 as the path tends to this parabolic Blaschke product. It turns out that the limiting parabolic Blaschke product on the unit circle is conjugate to the famous Boole map on the real line. Thus we can give a new explanation of Boole's formula discovered more than one hundred and fifty years ago. We modify the first smooth path to get a second smooth path in the space of expanding Blaschke products. The second smooth path tends to a totally degenerate map. We see that the first and second smooth paths have completely different asymptotic behaviors near the boundary of the space of expanding Blaschke products. However, they represent the same smooth path in the space of all smooth conjugacy classes of expanding Blaschke products. We use this to give a complete description of the global graph of the metric entropy on the space of expanding Blaschke products. We prove that the global graph looks like a bell. It is the first result to show a global picture of the metric entropy on a space of hyperbolic dynamical systems. We apply our results to the measure-theoretic entropy of a quadratic polynomial with respect to its Gibbs measure on its Julia set. We prove that the measure-theoretic entropy on the main cardioid of the Mandelbrot set is a real analytic function and asymptotically zero near the boundary.

Keywords: Expanding Blaschke product, parabolic Blaschke product, metric entropy, global graph, the main cardioid of the Mandelbrot set, harmonic measure, Gibbs measure, bell.
Mathematics Subject Classification: Primary: 37A05, 37A35; Secondary: 37F15, 37A30.
Citation: Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

R. Adler and B. Weiss, Ergodic infinite measure preserving transformation of Boole, Israel Journal of Mathematics, 16 (1973), 263-278.  doi: 10.1007/BF02756706.  Google Scholar

[3]

G. Boole, On the comparison of transcendents with certain applications to the theory of definite integrals, Philos. Trans. Roy. Soc. London, 147, Part III (1857), 745–803. doi: 10.1098/rstl.1857.0037.  Google Scholar

[4]

C. Doering and R. Mañé, The dynamics of Inner Functions, Ensaios Matemáticos, 3 (1991), 2-80.   Google Scholar

[5] A. Katok and B. Hasselbratt, Introduction to the Modern Theorey of Dynamical Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511809187.  Google Scholar
[6]

H. Hu, M. Jiang and Y. Jiang, Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure, Discrete and Continuous Dynamical Systems, 22 (2008), 215-234.  doi: 10.3934/dcds.2008.22.215.  Google Scholar

[7]

H. Hu, M. Jiang and Y. Jiang, Infimum of the metric entropy of volume preserving Anosov systems, Discrete and Continuous Dynamical Systems, 37 (2017), 4767-4783.  doi: 10.3934/dcds.2017205.  Google Scholar

[8]

Y. Jiang, Symmetric invariant measures, Contemporary Mathematics, AMS, Vol. 575, 2012,211–218.  Google Scholar

[9]

Y. Jiang, Geometric Gibbs theory, Science China–Mathematics, online first June 25, 2020. doi: 10.1007/s11425-019-1638-6.  Google Scholar

[10]

R. Mañè, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin-Heidelberg, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[11]

G. Pianigiani, First return map and invariant measures, Israel Journal of Mathematics, 35 (1980), 32-48.  doi: 10.1007/BF02760937.  Google Scholar

[12]

F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, Proc. ICM 2018, vol. 2, 2081–2160. Google Scholar

[13]

F. Przytycki, M. Urbański and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I, Annals of Mathematics, 130 (1989), 1-40.  doi: 10.2307/1971475.  Google Scholar

[14]

F. Przytycki, M. Urbański and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II, Studia Mathematica, 97 (1991), 189-225.  doi: 10.4064/sm-97-3-189-225.  Google Scholar

[15]

D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134.  Google Scholar

[16]

M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.  doi: 10.2307/2373276.  Google Scholar

[17]

M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Erg. Th. & Dyn. Sys., 5 (1985), 285-289.  doi: 10.1017/S014338570000290X.  Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

R. Adler and B. Weiss, Ergodic infinite measure preserving transformation of Boole, Israel Journal of Mathematics, 16 (1973), 263-278.  doi: 10.1007/BF02756706.  Google Scholar

[3]

G. Boole, On the comparison of transcendents with certain applications to the theory of definite integrals, Philos. Trans. Roy. Soc. London, 147, Part III (1857), 745–803. doi: 10.1098/rstl.1857.0037.  Google Scholar

[4]

C. Doering and R. Mañé, The dynamics of Inner Functions, Ensaios Matemáticos, 3 (1991), 2-80.   Google Scholar

[5] A. Katok and B. Hasselbratt, Introduction to the Modern Theorey of Dynamical Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511809187.  Google Scholar
[6]

H. Hu, M. Jiang and Y. Jiang, Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure, Discrete and Continuous Dynamical Systems, 22 (2008), 215-234.  doi: 10.3934/dcds.2008.22.215.  Google Scholar

[7]

H. Hu, M. Jiang and Y. Jiang, Infimum of the metric entropy of volume preserving Anosov systems, Discrete and Continuous Dynamical Systems, 37 (2017), 4767-4783.  doi: 10.3934/dcds.2017205.  Google Scholar

[8]

Y. Jiang, Symmetric invariant measures, Contemporary Mathematics, AMS, Vol. 575, 2012,211–218.  Google Scholar

[9]

Y. Jiang, Geometric Gibbs theory, Science China–Mathematics, online first June 25, 2020. doi: 10.1007/s11425-019-1638-6.  Google Scholar

[10]

R. Mañè, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin-Heidelberg, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[11]

G. Pianigiani, First return map and invariant measures, Israel Journal of Mathematics, 35 (1980), 32-48.  doi: 10.1007/BF02760937.  Google Scholar

[12]

F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, Proc. ICM 2018, vol. 2, 2081–2160. Google Scholar

[13]

F. Przytycki, M. Urbański and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I, Annals of Mathematics, 130 (1989), 1-40.  doi: 10.2307/1971475.  Google Scholar

[14]

F. Przytycki, M. Urbański and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II, Studia Mathematica, 97 (1991), 189-225.  doi: 10.4064/sm-97-3-189-225.  Google Scholar

[15]

D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134.  Google Scholar

[16]

M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.  doi: 10.2307/2373276.  Google Scholar

[17]

M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Erg. Th. & Dyn. Sys., 5 (1985), 285-289.  doi: 10.1017/S014338570000290X.  Google Scholar

Figure 1.  The global graph of the metric entropy $ \mathcal E $ on $ S \mathcal{B} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  A graph of the metric entropy $ {\mathcal E}_{d} $ on $ S \mathcal{B}(d) $ for $ d>2 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Xi Chen, Zongrun Wang, Songhai Deng, Yong Fang. Risk measure optimization: Perceived risk and overconfidence of structured product investors. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1473-1492. doi: 10.3934/jimo.2018105
[2]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
[3]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
[4]

Anja Randecker, Giulio Tiozzo. Cusp excursion in hyperbolic manifolds and singularity of harmonic measure. Journal of Modern Dynamics, 2021, 17: 183-211. doi: 10.3934/jmd.2021006
[5]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076
[6]

Canela Jordi. Singular perturbations of Blaschke products and connectivity of Fatou components. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3567-3585. doi: 10.3934/dcds.2017153
[7]

Haifeng Chu. Surgery on Herman rings of the standard Blaschke family. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 63-74. doi: 10.3934/dcds.2018003
[8]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[9]

Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287-296. doi: 10.3934/proc.2015.0287
[10]

Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1
[11]

Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159
[12]

François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
[13]

Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673
[14]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327
[15]

Bang-Xian Han. New characterizations of Ricci curvature on RCD metric measure spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4915-4927. doi: 10.3934/dcds.2018214
[16]

Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
[17]

Shengliang Pan, Deyan Zhang, Zhongjun Chao. A generalization of the Blaschke-Lebesgue problem to a kind of convex domains. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1587-1601. doi: 10.3934/dcdsb.2016012
[18]

Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315
[19]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[20]

Manfred Einsiedler, Elon Lindenstrauss. Symmetry of entropy in higher rank diagonalizable actions and measure classification. Journal of Modern Dynamics, 2018, 13: 163-185. doi: 10.3934/jmd.2018016

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (199)
  • HTML views (212)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy