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2012, 6(2): 183-203. doi: 10.3934/jmd.2012.6.183

Forty years of unimodal dynamics: On the occasion of Artur Avila winning the Brin Prize

1. 

Mathematics Department, Stony Brook University, Stony Brook, NY, 11794-3651, USA Government

Published  August 2012

The field of one-dimensional dynamics, real and complex, emerged from obscurity in the 1970s and has been intensely explored ever since. It combines the depth and complexity of chaotic phenomena with a chance to fully understand it in probabilistic terms: to describe the dynamics of typical orbits for typical maps. It also revealed fascinating universality features that had never been noticed before. The interplay between real and complex worlds illuminated by beautiful pictures of fractal structures adds special charm to the field. By now, we have reached a full probabilistic understanding of real analytic unimodal dynamics, and Artur Avila has been the key player in the final stage of the story (which roughly started with the new century). To put his work into perspective, we will begin with an overview of the main events in the field from the 1970s up to the end of the last century. Then we will describe Avila's work on unimodal dynamics that effectively closed up the field. We will finish by describing his results in the closely related direction, the geometry of Feigenbaum Julia sets, including a recent construction of a new class of Julia sets of positive area.
Citation: Mikhail Lyubich. Forty years of unimodal dynamics: On the occasion of Artur Avila winning the Brin Prize. Journal of Modern Dynamics, 2012, 6 (2) : 183-203. doi: 10.3934/jmd.2012.6.183
References:
[1]

A. Avila and M. Lyubich, Examples of Feigenbaum Julia sets with small Hausdorff dimension,, in, (2006), 71.

[2]

A. Avila and M. Lyubich, Hausdorff dimension and conformal measures of Feigenbaum Julia sets,, J. of the AMS, 21 (2008), 305.

[3]

A. Avila and M. Lyubich, The full renormalization horseshoe for unimodal maps of higher degree: Exponential contraction along hybrid classes,, Publ. Math. IHÉS, 114 (2011), 171.

[4]

A. Avila and M. Lyubich, Feigenbaum Julia sets of positive area,, Manuscript, (2011).

[5]

A. Avila, M. Lyubich and W. de Melo, Regular or stochastic dynamics in real analytic families of unimodal maps,, Invent. Math., 154 (2003), 451. doi: 10.1007/s00222-003-0307-6.

[6]

A. Avila, J. Kahn, M. Lyubich and W. Shen, Combinatorial rigidity for unicritical polynomials,, Annals of Math. (2), 170 (2009), 783. doi: 10.4007/annals.2009.170.783.

[7]

A. Avila, M. Lyubich and W. Shen, Parapuzzle of the Multibrot set and typical dynamics of unimodal maps,, J. of European Math. Soc., 13 (2011), 27. doi: 10.4171/JEMS/243.

[8]

A. Avila and C. G. Moreira, Statistical properties ofunimodal maps: The quadratic family,, Annals of Math. (2), 161 (2005), 831.

[9]

A. Avila and C. G. Moreira, Statistical properties of unimodal maps: Smooth families with negative Schwarzian derivative,, in, 286 (2003), 81.

[10]

A. Avila and C. G. Moreira, Statistical properties of unimodal maps: Physical measures, periodic orbits and pathological laminations,, Publications Math. IHÉS, 101 (2005), 1.

[11]

A. Avila and C. G. Moreira, Hausdorff dimension and the quadratic family,, Manuscript, (2002).

[12]

M. Benedicks and L. Carleson, On iterations of $1-ax^2$ on (-1,1),, Annals of Math. (2), 122 (1985), 1. doi: 10.2307/1971367.

[13]

M. Benedicks and L. Carleson, The dynamics of the Hénon map,, Annals of Math. (2), 133 (1991), 73. doi: 10.2307/2944326.

[14]

X. Buff and A. Cheritat, Quadratic Julia sets with positive area (2008),, , ().

[15]

B. Branner and J. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns,, Acta Math., 169 (1992), 229.

[16]

R. Brooks and J. Matelski, The dynamics of 2-generator subgroups of $\PSL(2, \C)$,, in, (1978), 65.

[17]

A. Blokh and M. Lyubich, Measurable dynamics of $S$-unimodal maps of the interval,, Ann. Sci. Éc. Norm Sup. (4), 24 (1991), 545.

[18]

H. Bruin, G. Keller, T. Nowicki and S. van Strien, Wild Cantor attractors exist,, Annals of Math. (2), 143 (1996), 97. doi: 10.2307/2118654.

[19]

H. Bruin, W. Shen and S. van Strien, Existence of unique SRB-measures is typical for real unicritical polynomial families,, Ann. Sci. Éc. Norm. Sup. (4), 39 (2006), 381.

[20]

L. Carleson and T. Gamelin, "Complex Dynamics,", Universitext: Tracts in Mathematics, (1993).

[21]

T. Clark, "Real and Complex Dynamics of Unicritical Maps,", Ph.D. Thesis,, University of Toronto, (2010).

[22]

P. Collet and J.-P. Eckmann, "Iterated Maps of the Interval as Dynamical Systems,", Progress in Physics, 1 (1980).

[23]

P. Collet and J.-P. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval,, Erg. Th. and Dyn Syst., 3 (1983), 13.

[24]

D. Cheragni, "Dynamics of Complex Unicritical Polynomials,", Ph.D. Thesis, (2009).

[25]

A. Douady, Description of compact sets in $\C$,, in, (1993), 429.

[26]

A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes,, preprint, (): 84.

[27]

A. Douady and J. H. Hubbard, On the dynamics of polynomial-like maps,, Ann. Sc. Éc. Norm. Sup. (4), 18 (1985), 287.

[28]

H. Epstein, Fixed points of the period-doubling operator,, Lecture notes, (1992).

[29]

E. de Faria, W. de Melo and A. Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings,, Annals of Math. (2), 164 (2006), 731. doi: 10.4007/annals.2006.164.731.

[30]

J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps,, Comm. Math. Physics., 70 (1979), 133. doi: 10.1007/BF01982351.

[31]

J. Graczyk and G. Światek, Generic hyperbolicity in the logistic family,, Annals of Math. (2), 146 (1997), 1.

[32]

H. Inou and M. Shishikura, The renormalization for parabolic fixed points and their perturbations,, Manuscript, (2006).

[33]

J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: Three theorems of J.-C. Yoccoz,, in, (1993), 467.

[34]

J. Hu and Y. Jiang, The Julia set of the Feigenbaum quadratic polynomial is locally connected,, Manuscript, (1993).

[35]

F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure,, Comm. Math. Phys., 127 (1990), 319.

[36]

M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps,, Comm. Math. Phys., 81 (1981), 39. doi: 10.1007/BF01941800.

[37]

S. Johnson, Singular measures without restrictive intervals,, Comm. Math. Phys., 110 (1987), 185. doi: 10.1007/BF01207362.

[38]

L. Jonker and D. Rand, Bifurcations in one dimension. I. The nonwandering set,, Inventiones Math., 62 (1981), 347.

[39]

J. Kahn, A priori bounds for some infinitely renormalizable quadratics. I. Bounded primitive combinatorics,, preprint, (2006).

[40]

J. Kahn and M. Lyubich, The quasi-additivity Law in conformal geometry,, Annals of Math. (2), 169 (2009), 561. doi: 10.4007/annals.2009.169.561.

[41]

J. Kahn and M. Lyubich, Local connectivity of Julia sets for unictritical polynomials,, Annals of Math. (2), 170 (2009), 413. doi: 10.4007/annals.2009.170.783.

[42]

J. Kahn and M. Lyubich, A priori bounds for some infinitely renormalizable quadratics. II. Decorations,, Annals Sci. École Norm. Sup. (4), 41 (2008), 57.

[43]

O. Kozlovski, Axiom A maps are dense in the space of unimodal maps in the $C^k$ topology,, Annals of Math. (2), 157 (2003), 1. doi: 10.4007/annals.2003.157.1.

[44]

O. Kozlovski, W. Shen and S. van Strien, Rigidity for real polynomials,, Annals of Math. (2), 165 (2007), 749. doi: 10.4007/annals.2007.165.749.

[45]

O. Kozlovski, W. Shen and S. van Strien, Density of hyperbolicity in dimension one,, Annals of Math. (2), 166 (2007), 145. doi: 10.4007/annals.2007.166.145.

[46]

G. Levin and S. van Strien, Local connectivity of the Julia set of real polynomials,, Annals of Math. (2), 147 (1998), 471. doi: 10.2307/120958.

[47]

O. E. Lanford III, A computer assisted proof of the Feigenbaum conjectures,, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427.

[48]

F. Ledrappier, Some properties of an absolutely continuous invariant measure on an interval,, Erg. Th. and Dyn. Syst., 1 (1981), 77.

[49]

Y. Lyubich, "Introduction to the Theory of Banach Representations of Groups,", Birkhäuser, (1988).

[50]

M. Lyubich, On the Lebesgue measure of the Julia set of a quadratic polynomial,, preprint, (1991).

[51]

M. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps,, Annals of Math. (2), 140 (1994), 347. doi: 10.2307/2118604.

[52]

M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math.,, 178 (1997), 178 (1997), 185.

[53]

M. Lyubich, How big is the set of infinitely renormalizable quadratics?,, in, 184 (1998), 131.

[54]

M. Lyubich, Dynamics of quadratic polynomials. III. Parapuzzle and SBR measures,, Astérisque, 261 (2000), 173.

[55]

M. Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture,, Annals of Math. (2), 149 (1999), 319. doi: 10.2307/120968.

[56]

M. Lyubich, Almost every real quadratic map is either regular or stochastic,, Annals of Math. (2), 156 (2002), 1. doi: 10.2307/3597183.

[57]

M. Lyubich and J. Milnor, The Fibonacci unimodal map,, Journal of AMS, 6 (1993), 425.

[58]

M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: Complex bounds for real maps,, Ann. Inst. Fourier (Grenoble), 47 (1997), 1219. doi: 10.5802/aif.1598.

[59]

T. Y. Li and J. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254.

[60]

M. Martens, Distortion results and invariant Cantor sets of unimodal maps,, Erg. Th. & Dyn. Syst., 14 (1994), 331.

[61]

M. Martens, The periodic points of renormalization,, Ann. Math., 147 (1998), 543. doi: 10.2307/120959.

[62]

M. Martens and T. Nowicki, Invariant measures for Lebesgue typical quadratic maps,, Astérisque, 261 (2000), 239.

[63]

M. Martens and W. de Melo, The multipliers of periodic points in one-dimensional dynamics,, Nonlinearity, 12 (1999), 217. doi: 10.1088/0951-7715/12/2/003.

[64]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459. doi: 10.1038/261459a0.

[65]

C. McMullen, "Complex Dynamics and Renormalization,", Princeton University Press, (1994).

[66]

C. McMullen, "Renormalization and Three Manifolds which Fiber Over the Circle,", Princeton University Press, (1996).

[67]

C. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets,, Acta Math., 180 (1998), 247. doi: 10.1007/BF02392901.

[68]

M. Misiurewicz, Absolutely continuous measures for certain maps of an interval,, IHÉS Publ. Math., 53 (1981), 17.

[69]

M. Metropolis, M. Stein and P. Stein, On finite limit sets for transformations on the unit interval,, J. Combinatorial Theory Ser. A, 15 (1973), 25. doi: 10.1016/0097-3165(73)90033-2.

[70]

W. de Melo and S. van Strien, "One-Dimensional Dynamics,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25 (1993).

[71]

J. Milnor, "Dynamics in One Complex Variable,", Third edition, 160 (2006).

[72]

J. Milnor, On the concept of attractor,, Comm. Math. Physics, 99 (1985), 177. doi: 10.1007/BF01212280.

[73]

J. Milnor, Local connectivity of Julia sets: Expository lectures,, in, 274 (2000), 67.

[74]

J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory,, Math. Intell., 19 (1997), 30. doi: 10.1007/BF03024428.

[75]

J. Milnor and W. Thurston, On iterated maps of the interval,, in, 1342 (1988), 1986.

[76]

P. J. Myrberg, Sur l'itération des polynomes réels quadratiques,, J. Math. Pures Appl. (9), 41 (1962), 339.

[77]

T. Nowicki and S. van Strien, Absolutely continuous measures for c2 unimodal maps satisfying the Collet-Eckmann conditions,, Invent. Math., 93 (1988), 619. doi: 10.1007/BF01410202.

[78]

J. Palis, A global view of dynamics and a Conjecture of the denseness of finitude of attractors,, Astérique, 261 (2000), 335.

[79]

E. A. Prado, Ergodicity of conformal measures for unimodal polynomials,, Conform. Geom. Dyn., 2 (1998), 29.

[80]

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[81]

D. Sullivan, Quasiconformal homeomorphisms and dynamics, topology, and geometry,, in, (1986), 1216.

[82]

D. Sullivan, Bounds, quadratic differentials, and renormalization conjetures,, in, (1988).

[83]

W. Shen, Decay of geometry for unimodal maps: An elementary proof,, Annals of Math. (2), 163 (2006), 383. doi: 10.4007/annals.2006.163.383.

[84]

D. Smania, On the hyperbolicity of the period-doubling fixed point,, Trans. AMS, 358 (2006), 1827.

[85]

M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets,, Annals of Math. (2), 147 (1998), 225. doi: 10.2307/121009.

[86]

M. Shishikura, Topological, geometric and complex analytic properties of Julia sets,, in, (1995), 886.

[87]

M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents,, Invetiones Math., 139 (2000), 495. doi: 10.1007/s002229900035.

[88]

W. Thurston, On the geometry and dynamics of iterated rational maps,, in, (2009), 3.

[89]

E. B. Vul, Y. G. Sinaĭ and K. M. Khanin, Feigenbaum universality and the thermodynamical formalism,, Russian Math. Surveys, 39 (1984), 1.

[90]

M. Yampolsky, Siegel disks and renormalization fixed points,, in, 53 (2008), 377.

[91]

B. Yarrington, "Local Connectivity and Lebesgue Measure of Polynomial Julia Sets,", Ph.D. Thesis, (1995).

[92]

L.-S. Young, Decay of correlations for certain quadratic maps,, Comm. Math. Phys., 146 (1992), 123. doi: 10.1007/BF02099211.

show all references

References:
[1]

A. Avila and M. Lyubich, Examples of Feigenbaum Julia sets with small Hausdorff dimension,, in, (2006), 71.

[2]

A. Avila and M. Lyubich, Hausdorff dimension and conformal measures of Feigenbaum Julia sets,, J. of the AMS, 21 (2008), 305.

[3]

A. Avila and M. Lyubich, The full renormalization horseshoe for unimodal maps of higher degree: Exponential contraction along hybrid classes,, Publ. Math. IHÉS, 114 (2011), 171.

[4]

A. Avila and M. Lyubich, Feigenbaum Julia sets of positive area,, Manuscript, (2011).

[5]

A. Avila, M. Lyubich and W. de Melo, Regular or stochastic dynamics in real analytic families of unimodal maps,, Invent. Math., 154 (2003), 451. doi: 10.1007/s00222-003-0307-6.

[6]

A. Avila, J. Kahn, M. Lyubich and W. Shen, Combinatorial rigidity for unicritical polynomials,, Annals of Math. (2), 170 (2009), 783. doi: 10.4007/annals.2009.170.783.

[7]

A. Avila, M. Lyubich and W. Shen, Parapuzzle of the Multibrot set and typical dynamics of unimodal maps,, J. of European Math. Soc., 13 (2011), 27. doi: 10.4171/JEMS/243.

[8]

A. Avila and C. G. Moreira, Statistical properties ofunimodal maps: The quadratic family,, Annals of Math. (2), 161 (2005), 831.

[9]

A. Avila and C. G. Moreira, Statistical properties of unimodal maps: Smooth families with negative Schwarzian derivative,, in, 286 (2003), 81.

[10]

A. Avila and C. G. Moreira, Statistical properties of unimodal maps: Physical measures, periodic orbits and pathological laminations,, Publications Math. IHÉS, 101 (2005), 1.

[11]

A. Avila and C. G. Moreira, Hausdorff dimension and the quadratic family,, Manuscript, (2002).

[12]

M. Benedicks and L. Carleson, On iterations of $1-ax^2$ on (-1,1),, Annals of Math. (2), 122 (1985), 1. doi: 10.2307/1971367.

[13]

M. Benedicks and L. Carleson, The dynamics of the Hénon map,, Annals of Math. (2), 133 (1991), 73. doi: 10.2307/2944326.

[14]

X. Buff and A. Cheritat, Quadratic Julia sets with positive area (2008),, , ().

[15]

B. Branner and J. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns,, Acta Math., 169 (1992), 229.

[16]

R. Brooks and J. Matelski, The dynamics of 2-generator subgroups of $\PSL(2, \C)$,, in, (1978), 65.

[17]

A. Blokh and M. Lyubich, Measurable dynamics of $S$-unimodal maps of the interval,, Ann. Sci. Éc. Norm Sup. (4), 24 (1991), 545.

[18]

H. Bruin, G. Keller, T. Nowicki and S. van Strien, Wild Cantor attractors exist,, Annals of Math. (2), 143 (1996), 97. doi: 10.2307/2118654.

[19]

H. Bruin, W. Shen and S. van Strien, Existence of unique SRB-measures is typical for real unicritical polynomial families,, Ann. Sci. Éc. Norm. Sup. (4), 39 (2006), 381.

[20]

L. Carleson and T. Gamelin, "Complex Dynamics,", Universitext: Tracts in Mathematics, (1993).

[21]

T. Clark, "Real and Complex Dynamics of Unicritical Maps,", Ph.D. Thesis,, University of Toronto, (2010).

[22]

P. Collet and J.-P. Eckmann, "Iterated Maps of the Interval as Dynamical Systems,", Progress in Physics, 1 (1980).

[23]

P. Collet and J.-P. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval,, Erg. Th. and Dyn Syst., 3 (1983), 13.

[24]

D. Cheragni, "Dynamics of Complex Unicritical Polynomials,", Ph.D. Thesis, (2009).

[25]

A. Douady, Description of compact sets in $\C$,, in, (1993), 429.

[26]

A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes,, preprint, (): 84.

[27]

A. Douady and J. H. Hubbard, On the dynamics of polynomial-like maps,, Ann. Sc. Éc. Norm. Sup. (4), 18 (1985), 287.

[28]

H. Epstein, Fixed points of the period-doubling operator,, Lecture notes, (1992).

[29]

E. de Faria, W. de Melo and A. Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings,, Annals of Math. (2), 164 (2006), 731. doi: 10.4007/annals.2006.164.731.

[30]

J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps,, Comm. Math. Physics., 70 (1979), 133. doi: 10.1007/BF01982351.

[31]

J. Graczyk and G. Światek, Generic hyperbolicity in the logistic family,, Annals of Math. (2), 146 (1997), 1.

[32]

H. Inou and M. Shishikura, The renormalization for parabolic fixed points and their perturbations,, Manuscript, (2006).

[33]

J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: Three theorems of J.-C. Yoccoz,, in, (1993), 467.

[34]

J. Hu and Y. Jiang, The Julia set of the Feigenbaum quadratic polynomial is locally connected,, Manuscript, (1993).

[35]

F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure,, Comm. Math. Phys., 127 (1990), 319.

[36]

M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps,, Comm. Math. Phys., 81 (1981), 39. doi: 10.1007/BF01941800.

[37]

S. Johnson, Singular measures without restrictive intervals,, Comm. Math. Phys., 110 (1987), 185. doi: 10.1007/BF01207362.

[38]

L. Jonker and D. Rand, Bifurcations in one dimension. I. The nonwandering set,, Inventiones Math., 62 (1981), 347.

[39]

J. Kahn, A priori bounds for some infinitely renormalizable quadratics. I. Bounded primitive combinatorics,, preprint, (2006).

[40]

J. Kahn and M. Lyubich, The quasi-additivity Law in conformal geometry,, Annals of Math. (2), 169 (2009), 561. doi: 10.4007/annals.2009.169.561.

[41]

J. Kahn and M. Lyubich, Local connectivity of Julia sets for unictritical polynomials,, Annals of Math. (2), 170 (2009), 413. doi: 10.4007/annals.2009.170.783.

[42]

J. Kahn and M. Lyubich, A priori bounds for some infinitely renormalizable quadratics. II. Decorations,, Annals Sci. École Norm. Sup. (4), 41 (2008), 57.

[43]

O. Kozlovski, Axiom A maps are dense in the space of unimodal maps in the $C^k$ topology,, Annals of Math. (2), 157 (2003), 1. doi: 10.4007/annals.2003.157.1.

[44]

O. Kozlovski, W. Shen and S. van Strien, Rigidity for real polynomials,, Annals of Math. (2), 165 (2007), 749. doi: 10.4007/annals.2007.165.749.

[45]

O. Kozlovski, W. Shen and S. van Strien, Density of hyperbolicity in dimension one,, Annals of Math. (2), 166 (2007), 145. doi: 10.4007/annals.2007.166.145.

[46]

G. Levin and S. van Strien, Local connectivity of the Julia set of real polynomials,, Annals of Math. (2), 147 (1998), 471. doi: 10.2307/120958.

[47]

O. E. Lanford III, A computer assisted proof of the Feigenbaum conjectures,, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427.

[48]

F. Ledrappier, Some properties of an absolutely continuous invariant measure on an interval,, Erg. Th. and Dyn. Syst., 1 (1981), 77.

[49]

Y. Lyubich, "Introduction to the Theory of Banach Representations of Groups,", Birkhäuser, (1988).

[50]

M. Lyubich, On the Lebesgue measure of the Julia set of a quadratic polynomial,, preprint, (1991).

[51]

M. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps,, Annals of Math. (2), 140 (1994), 347. doi: 10.2307/2118604.

[52]

M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math.,, 178 (1997), 178 (1997), 185.

[53]

M. Lyubich, How big is the set of infinitely renormalizable quadratics?,, in, 184 (1998), 131.

[54]

M. Lyubich, Dynamics of quadratic polynomials. III. Parapuzzle and SBR measures,, Astérisque, 261 (2000), 173.

[55]

M. Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture,, Annals of Math. (2), 149 (1999), 319. doi: 10.2307/120968.

[56]

M. Lyubich, Almost every real quadratic map is either regular or stochastic,, Annals of Math. (2), 156 (2002), 1. doi: 10.2307/3597183.

[57]

M. Lyubich and J. Milnor, The Fibonacci unimodal map,, Journal of AMS, 6 (1993), 425.

[58]

M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: Complex bounds for real maps,, Ann. Inst. Fourier (Grenoble), 47 (1997), 1219. doi: 10.5802/aif.1598.

[59]

T. Y. Li and J. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254.

[60]

M. Martens, Distortion results and invariant Cantor sets of unimodal maps,, Erg. Th. & Dyn. Syst., 14 (1994), 331.

[61]

M. Martens, The periodic points of renormalization,, Ann. Math., 147 (1998), 543. doi: 10.2307/120959.

[62]

M. Martens and T. Nowicki, Invariant measures for Lebesgue typical quadratic maps,, Astérisque, 261 (2000), 239.

[63]

M. Martens and W. de Melo, The multipliers of periodic points in one-dimensional dynamics,, Nonlinearity, 12 (1999), 217. doi: 10.1088/0951-7715/12/2/003.

[64]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459. doi: 10.1038/261459a0.

[65]

C. McMullen, "Complex Dynamics and Renormalization,", Princeton University Press, (1994).

[66]

C. McMullen, "Renormalization and Three Manifolds which Fiber Over the Circle,", Princeton University Press, (1996).

[67]

C. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets,, Acta Math., 180 (1998), 247. doi: 10.1007/BF02392901.

[68]

M. Misiurewicz, Absolutely continuous measures for certain maps of an interval,, IHÉS Publ. Math., 53 (1981), 17.

[69]

M. Metropolis, M. Stein and P. Stein, On finite limit sets for transformations on the unit interval,, J. Combinatorial Theory Ser. A, 15 (1973), 25. doi: 10.1016/0097-3165(73)90033-2.

[70]

W. de Melo and S. van Strien, "One-Dimensional Dynamics,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25 (1993).

[71]

J. Milnor, "Dynamics in One Complex Variable,", Third edition, 160 (2006).

[72]

J. Milnor, On the concept of attractor,, Comm. Math. Physics, 99 (1985), 177. doi: 10.1007/BF01212280.

[73]

J. Milnor, Local connectivity of Julia sets: Expository lectures,, in, 274 (2000), 67.

[74]

J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory,, Math. Intell., 19 (1997), 30. doi: 10.1007/BF03024428.

[75]

J. Milnor and W. Thurston, On iterated maps of the interval,, in, 1342 (1988), 1986.

[76]

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