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April  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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

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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. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[44]

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[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. Google Scholar

[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. Google Scholar

[47]

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

[48]

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

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

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

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

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

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

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

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

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

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

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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. Google Scholar

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

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

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show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

[56]

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

[57]

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

[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. Google Scholar

[59]

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

[60]

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

[61]

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

[62]

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

[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. Google Scholar

[64]

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

[65]

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

[66]

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

[67]

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

[68]

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