Commentary on “Differentiable dynamical systems” by Stephen Smale

Zworski, Maciej

doi:10.1090/bull/1618

Commentary on “Differentiable dynamical systems” by Stephen Smale
HTML articles powered by AMS MathViewer

by Maciej Zworski PDF

Bull. Amer. Math. Soc. 55 (2018), 331-336 Request permission

Abstract:

Immediately following the commentary below, this previously published article is reprinted in its entirety: Stephen Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc. 73 (1967), no. 6, 747–817.

References

Viviane Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 68, Springer Verlag, 2018.
Nguyen Viet Dang and Gabriel Rivière, Pollicott-Ruelle spectrum and Witten Laplacians, arXiv:1709.04265.
Semyon Dyatlov and Colin Guillarmou, Pollicott-Ruelle resonances for open systems, Ann. Henri Poincaré 17 (2016), no. 11, 3089–3146. MR 3556517, DOI 10.1007/s00023-016-0491-8
Semyon Dyatlov and Colin Guillarmou, Dynamical zeta functions for Axiom A flows, Bull. Amer. Math. Soc. (N.S.) 55 (2018), no. 3, 337–342.
Semyon Dyatlov and Maciej Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 3, 543–577 (English, with English and French summaries). MR 3503826, DOI 10.24033/asens.2290
Semyon Dyatlov and Maciej Zworski, Ruelle zeta function at zero for surfaces, Invent. Math. 210 (2017), no. 1, 211–229. MR 3698342, DOI 10.1007/s00222-017-0727-3
Frédéric Faure and Johannes Sjöstrand, Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys. 308 (2011), no. 2, 325–364 (English, with English and French summaries). MR 2851145, DOI 10.1007/s00220-011-1349-z
Frédéric Faure and Masato Tsujii, The semiclassical zeta function for geodesic flows on negatively curved manifolds, Invent. Math. 208 (2017), no. 3, 851–998. MR 3648976, DOI 10.1007/s00222-016-0701-5
David Fried, Meromorphic zeta functions for analytic flows, Comm. Math. Phys. 174 (1995), no. 1, 161–190. MR 1372805, DOI 10.1007/BF02099469
P. Giulietti, C. Liverani, and M. Pollicott, Anosov flows and dynamical zeta functions, Ann. of Math. (2) 178 (2013), no. 2, 687–773. MR 3071508, DOI 10.4007/annals.2013.178.2.6
Charles Hadfield, Zeta function at zero for surfaces with boundary, arXiv:1803.10982.
Richard B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) Lecture Notes in Pure and Appl. Math., vol. 161, Dekker, New York, 1994, pp. 85–130. MR 1291640
David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), no. 3, 231–242. MR 420720, DOI 10.1007/BF01403069
Hans Henrik Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 805–819. MR 1406435, DOI 10.1017/S0143385700009111
Shu Shen, Analytic torsion, dynamical zeta functions, and the Fried conjecture, Anal. PDE 11 (2018), no. 1, 1–74. MR 3707290, DOI 10.2140/apde.2018.11.1
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
András Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math. 194 (2013), no. 2, 381–513. MR 3117526, DOI 10.1007/s00222-012-0446-8