# American Institute of Mathematical Sciences

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

## Global graph of metric entropy on expanding Blaschke products

 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.

Citation: Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete and 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. [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. [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. [4] C. Doering and R. Mañé, The dynamics of Inner Functions, Ensaios Matemáticos, 3 (1991), 2-80. [5] A. Katok and B. Hasselbratt, Introduction to the Modern Theorey of Dynamical Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511809187. [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. [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. [8] Y. Jiang, Symmetric invariant measures, Contemporary Mathematics, AMS, Vol. 575, 2012,211–218. [9] Y. Jiang, Geometric Gibbs theory, Science China–Mathematics, online first June 25, 2020. doi: 10.1007/s11425-019-1638-6. [10] R. Mañè, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin-Heidelberg, 1987. doi: 10.1007/978-3-642-70335-5. [11] G. Pianigiani, First return map and invariant measures, Israel Journal of Mathematics, 35 (1980), 32-48.  doi: 10.1007/BF02760937. [12] F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, Proc. ICM 2018, vol. 2, 2081–2160. [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. [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. [15] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134. [16] M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.  doi: 10.2307/2373276. [17] M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Erg. Th. & Dyn. Sys., 5 (1985), 285-289.  doi: 10.1017/S014338570000290X.

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##### 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. [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. [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. [4] C. Doering and R. Mañé, The dynamics of Inner Functions, Ensaios Matemáticos, 3 (1991), 2-80. [5] A. Katok and B. Hasselbratt, Introduction to the Modern Theorey of Dynamical Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511809187. [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. [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. [8] Y. Jiang, Symmetric invariant measures, Contemporary Mathematics, AMS, Vol. 575, 2012,211–218. [9] Y. Jiang, Geometric Gibbs theory, Science China–Mathematics, online first June 25, 2020. doi: 10.1007/s11425-019-1638-6. [10] R. Mañè, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin-Heidelberg, 1987. doi: 10.1007/978-3-642-70335-5. [11] G. Pianigiani, First return map and invariant measures, Israel Journal of Mathematics, 35 (1980), 32-48.  doi: 10.1007/BF02760937. [12] F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, Proc. ICM 2018, vol. 2, 2081–2160. [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. [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. [15] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134. [16] M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.  doi: 10.2307/2373276. [17] M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Erg. Th. & Dyn. Sys., 5 (1985), 285-289.  doi: 10.1017/S014338570000290X.
The global graph of the metric entropy $\mathcal E$ on $S \mathcal{B}$
A graph of the metric entropy ${\mathcal E}_{d}$ on $S \mathcal{B}(d)$ for $d>2$
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