Abstract
Here we review some recent results that give a rather complete description of the dynamics of almost all mappings in real analytic families of unimodal maps
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A. Avila, M. Lyubich andW. de Melo,Regular or stochastic dynamics in real analytic families of unimodal maps, Inventiones Mathematicae n. 3,154 (2003), 451–550.
A. Avila and C. G. Moreira,Phase-Parameter relation and sharp statistical properties in general families of unimodal maps. http://www.math.sunysb.edu/artur/
L. Bers andH.L. Royden,Holomorphic families of injections, Acta Math.157 (1986), 259–286.
A. Douady andJ.H. Hubbard,On the dynamics of polynomial-like maps, Ann. Sc. Éc. Norm. Sup.18 (1985), 287–343.
J. Guckenheimer,Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Physics.70 (1979), 133–160.
J. Graczyk, D. Sands andG. Swiatek,Schwarzian derivative in unimodal dynamics, C.R. Acad/Sci. Paris332 (2001), no. 4, 329–332.
F. Hofbauer andG. Keller,Quadratic maps without asymptotic measure, Comm. Math. Physics127 (1990), 319–337.
M. Jacobson,Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys.81 (1981), 39–88.
G. Keller,Exponents, attractors, and Hopf decompositions for interval maps, Erg. Th. & Dyn. Syst.10 (1990), 717–744.
O.S. Kozlovsky,Axiom A maps are dense in the space of unimodal maps in the C k topology, Ann. of Math. (2)157, no. 1 (2003), 1–43.
O.S. Kozlovsky,Getting Rid of the negative Schwarzian derivative condition, Ann. Math.152 (2000), 743–762.
G. Levin andS. van Strien,Local connectivity of Julia sets of real polynomials, Ann. of Math.147 (1998), 471–541.
M. Lyubich,Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. Math.140 (1994), 347–404.
M. Lyubich,Dynamics of quadratic polynomials, I–II, Acta Math.178 (1997), 185–297.
M. Lyubich,Dynamics of quadratic polynomials, III. Parapuzzle and SBR measure, Asterisque261 (2000), 173–200.
M. Lyubich,Feigenbaum-Coullet-Tresser Universality, and Milnor's Hairiness Conjecture, Ann. Math.149 (1999), 319–420.
M. Lyubich andM. Yampolsky,Dynamics of quadratic polynomials: Complex bounds for real maps, Ann. Inst. Fourier47 (1997), 1219–1255.
G. Levin andS. van Strien,Local connectivity of Julia sets of real polynomials, Annals of Math.147 (1998), 471–541.
R. Mañé,Hyperbolicity, sinks and measures for one-dimensional dynamics, Comm. Math. Phys.100 (1985), 495–524.
M. Misiurewicz,Absolutely continuous measures for certain maps of an interval, Publ. Math. I.H.E.S.53 (1981), 17–51.
J. Milnor andW. Thurston,On iterated maps of an interval I, II, Springer Lecture Notes in Math. vol 1342, (1988), 465–563.
M. Martens, W. de Melo andS. van Strien,Julia-Fatou-Sullivan theory for real one dimensional dynamics, Acta Math.168 (1992), 273–318.
R. Mañé, P. Sad andD. Sullivan,On the dynamics of rational maps, Ann. scient. Ec. Norm. Sup.16 (1983), 193–217.
M. Martens,Distortion results and invariant Cantor sets for unimodal maps, Erg. Th. & Dyn. Syst.14 (1994), 331–349.
M. Martens andT. Nowicki,Invariant measures for Lebesgue typical quadratic maps, Asterisque261 (2000), 239–252.
C. McMullen,Complex dynamics and renormalization, Annals of Math. Studies135, Princeton University Press, 1994.
C. McMullen,Renormalization and 3-manifolds which fiber over the circle, Annals of Math. Studies142, Princeton University Press, 1996.
W. de Melo andS. van Strien,A structure theorem in one-dimensional dynamics, Ann. Math.129 (1989), 519–546.
W. de Melo and S. van Strien,One-dimensional dynamics, Springer, 1993.
D. Singer,Stable orbits and bifurcations of maps of the interval, SIAM J. App. Math.35 (1978), 260–267.
D. Sands,Misiurewwixz maps are rare, Comm. Math. Phys.197, no. 1, (1998), 109–129.
W. Shen,Decay of Geometry for Unimodal maps: an elementary proof. http://www.maths.warwick.ac.uk/staff/wxshen.html
D. Sullivan,Bounds, quadratic differentials, and renormalization conjectures, AMS Centennial Publications2: Mathematics into Twenty-first Century (1992).
D. Sullivan andW. Thurston,Extending holomorphic motions, Acta Math.157 (1986), 243–257.
Z. Slodkowsky,Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc.111 (1991), 347–355.
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To Jorge Sottomayor on his 60th birthday.
This work has been partially supported by FAPERJ grants E-26/151.896/2000, E-26/151.189/2002.
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de Melo, W. Bifurcation of unimodal maps. Qual. Th. Dyn. Syst 4, 413–424 (2004). https://doi.org/10.1007/BF02970867
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DOI: https://doi.org/10.1007/BF02970867