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On the Brin Prize work of Artur Avila in Teichmüller dynamics and intervalexchange transformations
Forty years of unimodal dynamics: On the occasion of Artur Avila winning the Brin Prize (Brin Prize article)
1.  Mathematics Department, Stony Brook University, Stony Brook, NY, 117943651, USA Government 
References:
[1] 
A. Avila and M. Lyubich, Examples of Feigenbaum Julia sets with small Hausdorff dimension, in "Dynamics on the Riemann Sphere'' (eds. P. Hjorth and C. Peterson), European Math. Soc., Zürich, (2006), 7187. 
[2] 
A. Avila and M. Lyubich, Hausdorff dimension and conformal measures of Feigenbaum Julia sets, J. of the AMS, 21 (2008), 305383. 
[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), 171223. 
[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), 451550. doi: 10.1007/s0022200303076. 
[6] 
A. Avila, J. Kahn, M. Lyubich and W. Shen, Combinatorial rigidity for unicritical polynomials, Annals of Math. (2), 170 (2009), 783797. 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), 2756. 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), 831881. 
[9] 
A. Avila and C. G. Moreira, Statistical properties of unimodal maps: Smooth families with negative Schwarzian derivative, in "Geometric Methods in Dynamics. I,'' Astérisque, 286 (2003), 81118. 
[10] 
A. Avila and C. G. Moreira, Statistical properties of unimodal maps: Physical measures, periodic orbits and pathological laminations, Publications Math. IHÉS, 101 (2005), 167. 
[11] 
A. Avila and C. G. Moreira, Hausdorff dimension and the quadratic family, Manuscript, 2002. 
[12] 
M. Benedicks and L. Carleson, On iterations of $1ax^2$ on (1,1), Annals of Math. (2), 122 (1985), 125. doi: 10.2307/1971367. 
[13] 
M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Math. (2), 133 (1991), 73169. doi: 10.2307/2944326. 
[14] 
X. Buff and A. Cheritat, Quadratic Julia sets with positive area (2008), arXiv:math/0605514. 
[15] 
B. Branner and J. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math., 169 (1992), 229325. 
[16] 
R. Brooks and J. Matelski, The dynamics of 2generator subgroups of $\PSL(2, \C)$, in "Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference'' (eds. I. Kra and B. Maskit), Ann. Math. Studies, no. 97, Princeton Univ. Press, 6572. 
[17] 
A. Blokh and M. Lyubich, Measurable dynamics of $S$unimodal maps of the interval, Ann. Sci. Éc. Norm Sup. (4), 24 (1991), 545573. 
[18] 
H. Bruin, G. Keller, T. Nowicki and S. van Strien, Wild Cantor attractors exist, Annals of Math. (2), 143 (1996), 97130. doi: 10.2307/2118654. 
[19] 
H. Bruin, W. Shen and S. van Strien, Existence of unique SRBmeasures is typical for real unicritical polynomial families, Ann. Sci. Éc. Norm. Sup. (4), 39 (2006), 381414. 
[20] 
L. Carleson and T. Gamelin, "Complex Dynamics," Universitext: Tracts in Mathematics, SpringerVerlag, New York, 1993. 
[21] 
T. Clark, "Real and Complex Dynamics of Unicritical Maps,", Ph.D. Thesis, University of Toronto, Canada, 2010. 
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P. Collet and J.P. Eckmann, "Iterated Maps of the Interval as Dynamical Systems," Progress in Physics, 1, Birkhäuser, Boston, Mass., 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), 1346. 
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D. Cheragni, "Dynamics of Complex Unicritical Polynomials," Ph.D. Thesis, State University of New York at Stony Brook, 2009. 
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A. Douady, Description of compact sets in $\C$, in "Topological Methods in Modern Mathematics" (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, (1993), 429465. 
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A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes, preprint, Publ. Math. d'Orsay, 842 and 854. 
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A. Douady and J. H. Hubbard, On the dynamics of polynomiallike maps, Ann. Sc. Éc. Norm. Sup. (4), 18 (1985), 287343. 
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H. Epstein, Fixed points of the perioddoubling operator, Lecture notes, Lausanne, 1992. 
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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), 731824. doi: 10.4007/annals.2006.164.731. 
[30] 
J. Guckenheimer, Sensitive dependence to initial conditions for onedimensional maps, Comm. Math. Physics., 70 (1979), 133160. doi: 10.1007/BF01982351. 
[31] 
J. Graczyk and G. Światek, Generic hyperbolicity in the logistic family, Annals of Math. (2), 146 (1997), 152. 
[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 "Topological Methods in Modern Mathematics" (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, (1993), 467511. 
[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), 319337. 
[36] 
M. Jakobson, Absolutely continuous invariant measures for oneparameter families of onedimensional maps, Comm. Math. Phys., 81 (1981), 3988. doi: 10.1007/BF01941800. 
[37] 
S. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys., 110 (1987), 185190. doi: 10.1007/BF01207362. 
[38] 
L. Jonker and D. Rand, Bifurcations in one dimension. I. The nonwandering set, Inventiones Math., 62 (1981), 347365. 
[39] 
J. Kahn, A priori bounds for some infinitely renormalizable quadratics. I. Bounded primitive combinatorics, preprint, IMS at Stony Brook, (2006). 
[40] 
J. Kahn and M. Lyubich, The quasiadditivity Law in conformal geometry, Annals of Math. (2), 169 (2009), 561593. 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), 413426. 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), 5784. 
[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), 143. 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), 749841. 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), 145182. 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), 471541. 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), 427434. 
[48] 
F. Ledrappier, Some properties of an absolutely continuous invariant measure on an interval, Erg. Th. and Dyn. Syst., 1 (1981), 7793. 
[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, IMS at Stony Brook, no. 10, (1991). 
[51] 
M. Lyubich, Combinatorics, geometry and attractors of quasiquadratic maps, Annals of Math. (2), 140 (1994), 347404. doi: 10.2307/2118604. 
[52] 
M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math., 178 (1997), 185297. 
[53] 
M. Lyubich, How big is the set of infinitely renormalizable quadratics?, in "Voronezh Winter Mathematical Schools," Amer. Math. Soc. Transl. Ser. 2, 184, Amer. Math. Soc., Providence, RI, (1998), 131143. 
[54] 
M. Lyubich, Dynamics of quadratic polynomials. III. Parapuzzle and SBR measures, Astérisque, 261 (2000), 173200. 
[55] 
M. Lyubich, FeigenbaumCoulletTresser universality and Milnor's hairiness conjecture, Annals of Math. (2), 149 (1999), 319420. doi: 10.2307/120968. 
[56] 
M. Lyubich, Almost every real quadratic map is either regular or stochastic, Annals of Math. (2), 156 (2002), 178. doi: 10.2307/3597183. 
[57] 
M. Lyubich and J. Milnor, The Fibonacci unimodal map, Journal of AMS, 6 (1993), 425457. 
[58] 
M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: Complex bounds for real maps, Ann. Inst. Fourier (Grenoble), 47 (1997), 12191255. doi: 10.5802/aif.1598. 
[59] 
T. Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985992. doi: 10.2307/2318254. 
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M. Martens, Distortion results and invariant Cantor sets of unimodal maps, Erg. Th. & Dyn. Syst., 14 (1994), 331349. 
[61] 
M. Martens, The periodic points of renormalization, Ann. Math., 147 (1998), 543584. doi: 10.2307/120959. 
[62] 
M. Martens and T. Nowicki, Invariant measures for Lebesgue typical quadratic maps, Astérisque, 261 (2000), 239252. 
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M. Martens and W. de Melo, The multipliers of periodic points in onedimensional dynamics, Nonlinearity, 12 (1999), 217227. doi: 10.1088/09517715/12/2/003. 
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R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459466. doi: 10.1038/261459a0. 
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C. McMullen, "Complex Dynamics and Renormalization," Princeton University Press, 1994. 
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C. McMullen, "Renormalization and Three Manifolds which Fiber Over the Circle," Princeton University Press, 1996. 
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show all references
References:
[1] 
A. Avila and M. Lyubich, Examples of Feigenbaum Julia sets with small Hausdorff dimension, in "Dynamics on the Riemann Sphere'' (eds. P. Hjorth and C. Peterson), European Math. Soc., Zürich, (2006), 7187. 
[2] 
A. Avila and M. Lyubich, Hausdorff dimension and conformal measures of Feigenbaum Julia sets, J. of the AMS, 21 (2008), 305383. 
[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), 171223. 
[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), 451550. doi: 10.1007/s0022200303076. 
[6] 
A. Avila, J. Kahn, M. Lyubich and W. Shen, Combinatorial rigidity for unicritical polynomials, Annals of Math. (2), 170 (2009), 783797. 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), 2756. 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), 831881. 
[9] 
A. Avila and C. G. Moreira, Statistical properties of unimodal maps: Smooth families with negative Schwarzian derivative, in "Geometric Methods in Dynamics. I,'' Astérisque, 286 (2003), 81118. 
[10] 
A. Avila and C. G. Moreira, Statistical properties of unimodal maps: Physical measures, periodic orbits and pathological laminations, Publications Math. IHÉS, 101 (2005), 167. 
[11] 
A. Avila and C. G. Moreira, Hausdorff dimension and the quadratic family, Manuscript, 2002. 
[12] 
M. Benedicks and L. Carleson, On iterations of $1ax^2$ on (1,1), Annals of Math. (2), 122 (1985), 125. doi: 10.2307/1971367. 
[13] 
M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Math. (2), 133 (1991), 73169. doi: 10.2307/2944326. 
[14] 
X. Buff and A. Cheritat, Quadratic Julia sets with positive area (2008), arXiv:math/0605514. 
[15] 
B. Branner and J. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math., 169 (1992), 229325. 
[16] 
R. Brooks and J. Matelski, The dynamics of 2generator subgroups of $\PSL(2, \C)$, in "Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference'' (eds. I. Kra and B. Maskit), Ann. Math. Studies, no. 97, Princeton Univ. Press, 6572. 
[17] 
A. Blokh and M. Lyubich, Measurable dynamics of $S$unimodal maps of the interval, Ann. Sci. Éc. Norm Sup. (4), 24 (1991), 545573. 
[18] 
H. Bruin, G. Keller, T. Nowicki and S. van Strien, Wild Cantor attractors exist, Annals of Math. (2), 143 (1996), 97130. doi: 10.2307/2118654. 
[19] 
H. Bruin, W. Shen and S. van Strien, Existence of unique SRBmeasures is typical for real unicritical polynomial families, Ann. Sci. Éc. Norm. Sup. (4), 39 (2006), 381414. 
[20] 
L. Carleson and T. Gamelin, "Complex Dynamics," Universitext: Tracts in Mathematics, SpringerVerlag, New York, 1993. 
[21] 
T. Clark, "Real and Complex Dynamics of Unicritical Maps,", Ph.D. Thesis, University of Toronto, Canada, 2010. 
[22] 
P. Collet and J.P. Eckmann, "Iterated Maps of the Interval as Dynamical Systems," Progress in Physics, 1, Birkhäuser, Boston, Mass., 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), 1346. 
[24] 
D. Cheragni, "Dynamics of Complex Unicritical Polynomials," Ph.D. Thesis, State University of New York at Stony Brook, 2009. 
[25] 
A. Douady, Description of compact sets in $\C$, in "Topological Methods in Modern Mathematics" (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, (1993), 429465. 
[26] 
A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes, preprint, Publ. Math. d'Orsay, 842 and 854. 
[27] 
A. Douady and J. H. Hubbard, On the dynamics of polynomiallike maps, Ann. Sc. Éc. Norm. Sup. (4), 18 (1985), 287343. 
[28] 
H. Epstein, Fixed points of the perioddoubling operator, Lecture notes, Lausanne, 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), 731824. doi: 10.4007/annals.2006.164.731. 
[30] 
J. Guckenheimer, Sensitive dependence to initial conditions for onedimensional maps, Comm. Math. Physics., 70 (1979), 133160. doi: 10.1007/BF01982351. 
[31] 
J. Graczyk and G. Światek, Generic hyperbolicity in the logistic family, Annals of Math. (2), 146 (1997), 152. 
[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 "Topological Methods in Modern Mathematics" (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, (1993), 467511. 
[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), 319337. 
[36] 
M. Jakobson, Absolutely continuous invariant measures for oneparameter families of onedimensional maps, Comm. Math. Phys., 81 (1981), 3988. doi: 10.1007/BF01941800. 
[37] 
S. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys., 110 (1987), 185190. doi: 10.1007/BF01207362. 
[38] 
L. Jonker and D. Rand, Bifurcations in one dimension. I. The nonwandering set, Inventiones Math., 62 (1981), 347365. 
[39] 
J. Kahn, A priori bounds for some infinitely renormalizable quadratics. I. Bounded primitive combinatorics, preprint, IMS at Stony Brook, (2006). 
[40] 
J. Kahn and M. Lyubich, The quasiadditivity Law in conformal geometry, Annals of Math. (2), 169 (2009), 561593. 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), 413426. 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), 5784. 
[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), 143. 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), 749841. 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), 145182. 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), 471541. 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), 427434. 
[48] 
F. Ledrappier, Some properties of an absolutely continuous invariant measure on an interval, Erg. Th. and Dyn. Syst., 1 (1981), 7793. 
[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, IMS at Stony Brook, no. 10, (1991). 
[51] 
M. Lyubich, Combinatorics, geometry and attractors of quasiquadratic maps, Annals of Math. (2), 140 (1994), 347404. doi: 10.2307/2118604. 
[52] 
M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math., 178 (1997), 185297. 
[53] 
M. Lyubich, How big is the set of infinitely renormalizable quadratics?, in "Voronezh Winter Mathematical Schools," Amer. Math. Soc. Transl. Ser. 2, 184, Amer. Math. Soc., Providence, RI, (1998), 131143. 
[54] 
M. Lyubich, Dynamics of quadratic polynomials. III. Parapuzzle and SBR measures, Astérisque, 261 (2000), 173200. 
[55] 
M. Lyubich, FeigenbaumCoulletTresser universality and Milnor's hairiness conjecture, Annals of Math. (2), 149 (1999), 319420. doi: 10.2307/120968. 
[56] 
M. Lyubich, Almost every real quadratic map is either regular or stochastic, Annals of Math. (2), 156 (2002), 178. doi: 10.2307/3597183. 
[57] 
M. Lyubich and J. Milnor, The Fibonacci unimodal map, Journal of AMS, 6 (1993), 425457. 
[58] 
M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: Complex bounds for real maps, Ann. Inst. Fourier (Grenoble), 47 (1997), 12191255. doi: 10.5802/aif.1598. 
[59] 
T. Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985992. doi: 10.2307/2318254. 
[60] 
M. Martens, Distortion results and invariant Cantor sets of unimodal maps, Erg. Th. & Dyn. Syst., 14 (1994), 331349. 
[61] 
M. Martens, The periodic points of renormalization, Ann. Math., 147 (1998), 543584. doi: 10.2307/120959. 
[62] 
M. Martens and T. Nowicki, Invariant measures for Lebesgue typical quadratic maps, Astérisque, 261 (2000), 239252. 
[63] 
M. Martens and W. de Melo, The multipliers of periodic points in onedimensional dynamics, Nonlinearity, 12 (1999), 217227. doi: 10.1088/09517715/12/2/003. 
[64] 
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459466. 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] 
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