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  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 - A, 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]

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
[2]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[3]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266
[4]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004
[5]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477
[6]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017
[7]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404
[8]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050
[9]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011
[10]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
[11]

Ömer Arslan, Selçuk Kürşat İşleyen. A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021002
[12]

Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021004
[13]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003
[14]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041
[15]

Sugata Gangopadhyay, Constanza Riera, Pantelimon Stănică. Gowers $ U_2 $ norm as a measure of nonlinearity for Boolean functions and their generalizations. Advances in Mathematics of Communications, 2021, 15 (2) : 241-256. doi: 10.3934/amc.2020056
[16]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055
[17]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319
[18]

Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021017
[19]

Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020295
[20]

Hongyu Cheng, Shimin Wang. Response solutions to harmonic oscillators beyond multi–dimensional brjuno frequency. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020222

2019 Impact Factor: 1.338

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences