Thermodynamic formalism methods in the theory of iteration of mappings in dimension one, real and complex
Abstract
The XIII Annual Lecture dedicated to the memory of Professor Andrzej Lasota
Keywords
one-dimensional dynamics; geometric pressure; thermodynamic formalism; equilibrium states; Hausdorff dimension; Lyapunov exponents
References
1. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes in Math., 470, Springer-Verlag, Berlin-New York, 1975.
2. H. Bruin, J. Rivera-Letelier, W. Shen, and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math. 172 (2008), no.3, 509–533.
3. W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin, 1993.
4. M. Denker, F. Przytycki, and M. Urbański, On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 255–266.
5. N. Dobbs and M. Todd, Free energy and equilibrium states for families of interval maps, to appear in Mem. Amer. Math. Soc. Avaliable at arXiv:1512.09245.
6. K. Gelfert, F. Przytycki, and M. Rams, On the Lyapunov spectrum for rational maps, Math. Ann. 348 (2010), no. 4, 965–1004.
7. K. Gelfert, F. Przytycki, and M. Rams, Lyapunov spectrum for multimodal maps, Ergodic Theory Dynam. Systems 36 (2016), no. 5, 1441–1493.
8. H. Hedenmalm and I. Kayumov, On the Makarov law of the iterated logarithm, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2235–2248.
9. I. Inoquio-Renteria and J. Rivera-Letelier, A characterization of hyperbolic potentials of rational maps, Bull. Braz. Math. Soc. (N.S.) 43 (2012), no. 1, 99–127.
10. A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488.
11. G. Levin, F. Przytycki, and W. Shen, The Lyapunov exponent of holomorphic maps, Invent. Math. 205 (2016), no. 2, 363–382.
12. F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math. 80 (1985), no. 1, 161–179.
13. F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), no. 1, 309–317.
14. F. Przytycki, Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps, Fund. Math. 144 (1994), no. 3, 259–278.
15. F. Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), no. 5, 2081–2099.
16. F. Przytycki, Geometric pressure in real and complex 1-dimensional dynamics via trees of pre-images and via spanning sets, Monatsh. Math. 185 (2018), no. 1, 133–158.
17. F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, in: B. Sirakov et al. (Eds.), Proceedings of the International Congress of Mathematicians 2018, Rio de Janeiro, Vol. III, World Scientific, Singapore, 2018, pp. 2105–2132.
18. F. Przytycki, Metody formalizmu termodynamicznego w jednowymiarowej dynamice rzeczywistej i zespolonej, Wiad. Mat. 54 (2018), no. 1, 23–53.
19. F. Przytycki and J. Rivera-Letelier, Nice inducing schemes and the thermodynamics of rational maps, Comm. Math. Phys. 301 (2011), no. 3, 661–707.
20. F. Przytycki and J. Rivera-Letelier, Geometric pressure for multimodal maps of the interval, Mem. Amer. Math. Soc. 259 (2019), no. 1246, 81 pp.
21. F. Przytycki, J. Rivera-Letelier, and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003), no. 1, 29–63.
22. F. Przytycki, J. Rivera-Letelier, and S. Smirnov, Equality of pressures for rational functions, Ergodic Theory Dynam. Systems 24 (2004), no. 3, 891–914.
23 F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fund. Math. 155 (1998), no. 2, 189–199.
24. F. Przytycki and J. Skrzypczak, Convergence and pre-images of limit points for coding trees for iterations of holomorphic maps, Math. Ann. 290 (1991), no. 3, 425–440.
25. F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, Cambridge, 2010.
26. F. Przytycki, M. Urbański, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I, Ann. of Math. (2) 130 (1989), no. 1, 1–40.
27. F. Przytycki, M. Urbański, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II, Studia Math. 97 (1991), no. 3, 189–225.
28. D. Ruelle, Thermodynamic Formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publ. Co., London, 1978.
29. Ya.G. Sinai, Gibbs measures in ergodic theory, (Russian) Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64. Russ. Math. Surv. 27 (1972), no. 4, 21–69 (English).
30. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
31. A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649.
2. H. Bruin, J. Rivera-Letelier, W. Shen, and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math. 172 (2008), no.3, 509–533.
3. W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin, 1993.
4. M. Denker, F. Przytycki, and M. Urbański, On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 255–266.
5. N. Dobbs and M. Todd, Free energy and equilibrium states for families of interval maps, to appear in Mem. Amer. Math. Soc. Avaliable at arXiv:1512.09245.
6. K. Gelfert, F. Przytycki, and M. Rams, On the Lyapunov spectrum for rational maps, Math. Ann. 348 (2010), no. 4, 965–1004.
7. K. Gelfert, F. Przytycki, and M. Rams, Lyapunov spectrum for multimodal maps, Ergodic Theory Dynam. Systems 36 (2016), no. 5, 1441–1493.
8. H. Hedenmalm and I. Kayumov, On the Makarov law of the iterated logarithm, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2235–2248.
9. I. Inoquio-Renteria and J. Rivera-Letelier, A characterization of hyperbolic potentials of rational maps, Bull. Braz. Math. Soc. (N.S.) 43 (2012), no. 1, 99–127.
10. A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488.
11. G. Levin, F. Przytycki, and W. Shen, The Lyapunov exponent of holomorphic maps, Invent. Math. 205 (2016), no. 2, 363–382.
12. F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math. 80 (1985), no. 1, 161–179.
13. F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), no. 1, 309–317.
14. F. Przytycki, Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps, Fund. Math. 144 (1994), no. 3, 259–278.
15. F. Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), no. 5, 2081–2099.
16. F. Przytycki, Geometric pressure in real and complex 1-dimensional dynamics via trees of pre-images and via spanning sets, Monatsh. Math. 185 (2018), no. 1, 133–158.
17. F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, in: B. Sirakov et al. (Eds.), Proceedings of the International Congress of Mathematicians 2018, Rio de Janeiro, Vol. III, World Scientific, Singapore, 2018, pp. 2105–2132.
18. F. Przytycki, Metody formalizmu termodynamicznego w jednowymiarowej dynamice rzeczywistej i zespolonej, Wiad. Mat. 54 (2018), no. 1, 23–53.
19. F. Przytycki and J. Rivera-Letelier, Nice inducing schemes and the thermodynamics of rational maps, Comm. Math. Phys. 301 (2011), no. 3, 661–707.
20. F. Przytycki and J. Rivera-Letelier, Geometric pressure for multimodal maps of the interval, Mem. Amer. Math. Soc. 259 (2019), no. 1246, 81 pp.
21. F. Przytycki, J. Rivera-Letelier, and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003), no. 1, 29–63.
22. F. Przytycki, J. Rivera-Letelier, and S. Smirnov, Equality of pressures for rational functions, Ergodic Theory Dynam. Systems 24 (2004), no. 3, 891–914.
23 F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fund. Math. 155 (1998), no. 2, 189–199.
24. F. Przytycki and J. Skrzypczak, Convergence and pre-images of limit points for coding trees for iterations of holomorphic maps, Math. Ann. 290 (1991), no. 3, 425–440.
25. F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, Cambridge, 2010.
26. F. Przytycki, M. Urbański, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I, Ann. of Math. (2) 130 (1989), no. 1, 1–40.
27. F. Przytycki, M. Urbański, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II, Studia Math. 97 (1991), no. 3, 189–225.
28. D. Ruelle, Thermodynamic Formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publ. Co., London, 1978.
29. Ya.G. Sinai, Gibbs measures in ergodic theory, (Russian) Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64. Russ. Math. Surv. 27 (1972), no. 4, 21–69 (English).
30. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
31. A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649.
PrzytyckiF. (2020). Thermodynamic formalism methods in the theory of iteration of mappings in dimension one, real and complex. Annales Mathematicae Silesianae, 35(1), 1-20. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13471
Feliks Przytycki
feliksp@impan.pl
Instytut Matematyki, Polska Akademia Nauk Poland
https://orcid.org/0000-0002-5713-948X
Instytut Matematyki, Polska Akademia Nauk Poland
https://orcid.org/0000-0002-5713-948X
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