Metadata
Abstract
We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in , as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family. We also describe the limit of the corresponding family of stationary measures on .
References
[ACS83] Large coupling behaviour of the Lyapunov exponent for tight binding one-dimensional random systems, J. Phys. A, Math. Gen., Volume 16 (1983) no. 7, pp. 209-211 | MR 707372 | Zbl 0568.34043
[Alp87] An elementary account of Selberg’s lemma, Enseign. Math., Volume 33 (1987) no. 3-4, pp. 269-273 | MR 925989
[Bea95] The geometry of discrete groups, Springer, Graduate Texts in Mathematics, Volume 91 (1995), xii+337 pages (Corrected reprint of the 1983 original) | MR 1393195
[Ber90] Spectral theory and analytic geometry over non-Archimedean fields, American Mathematical Society, Mathematical Surveys and Monographs, Volume 33 (1990), x+169 pages | MR 1070709
[Ber09] A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Birkhäuser (Progress in Mathematics) Volume 269 (2009), pp. 49-67 | Article | MR 2641170
[BJ17] Tropical and non-Archimedean limits of degenerating families of volume forms, J. Éc. Polytech., Math., Volume 4 (2017), pp. 87-139 | Article | MR 3611100
[BL85] Products of random matrices with applications to Schrödinger operators, Birkhäuser, Progress in Probability and Statistics, Volume 8 (1985), xii+283 pages | Article | MR 886674
[BNV17] Continuity of Lyapunov exponents for random two-dimensional matrices, Ergodic Theory Dyn. Syst., Volume 37 (2017) no. 5, pp. 1413-1442 | Article | MR 3667994
[BR10] Potential theory and dynamics on the Berkovich projective line, American Mathematical Society, Mathematical Surveys and Monographs, Volume 159 (2010), xxxiv+428 pages | Article | MR 2599526
[CM87] Group actions on -trees, Proc. Lond. Math. Soc., Volume 55 (1987) no. 3, pp. 571-604 | Article | MR 907233
[CS83] Varieties of group representations and splittings of -manifolds, Ann. Math., Volume 117 (1983) no. 1, pp. 109-146 | Article | MR 683804
[DD12] Random walks, Kleinian groups, and bifurcation currents, Invent. Math., Volume 190 (2012) no. 1, pp. 57-118 | Article | MR 2969274
[DD15] Lyapunov exponents for surface group representations, Commun. Math. Phys., Volume 340 (2015) no. 2, pp. 433-469 | Article | MR 3397023
[DeM03] Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann., Volume 326 (2003) no. 1, pp. 43-73 | Article | MR 1981611
[DeM16] Bifurcations, intersections, and heights, Algebra Number Theory, Volume 10 (2016) no. 5, pp. 1031-1056 | Article | MR 3531361
[DF16] Degenerations of complex dynamical systems II: analytic and algebraic stability, Math. Ann., Volume 365 (2016) no. 3-4, pp. 1669-1699 (With an appendix by Jan Kiwi) | Article | MR 3521102
[DMF14] Degenerations of complex dynamical systems, Forum Math. Sigma, Volume 2 (2014), e6, 36 pages | Article | | MR 3264250 | Zbl 1308.37023
[Fan18] Normalized Berkovich spaces and surface singularities, Trans. Am. Math. Soc., Volume 370 (2018) no. 11, pp. 7815-7859 | Article | MR 3852450
[Fav19] Degeneration of endomorphisms of the complex projective space in the hybrid space. (2019) (to appear in J. Inst. Math. Jussieu) | Article
[FJ04] The valuative tree, Springer, Lecture Notes in Mathematics, Volume 1853 (2004), xiv+234 pages | Article | MR 2097722
[FK83] Random matrix products and measures on projective spaces, Isr. J. Math., Volume 46 (1983) no. 1-2, pp. 12-32 | Article | MR 727020 | Zbl 0528.60028
[Fur63] Noncommuting random products, Trans. Am. Math. Soc., Volume 108 (1963), pp. 377-428 | Article | MR 0163345
[Fur02] Random walks on groups and random transformations, Handbook of dynamical systems, Vol. 1A, North-Holland (2002), pp. 931-1014 | Article | MR 1928529
[GR85] Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 69 (1985) no. 2, pp. 187-242 | Article | MR 779457 | Zbl 0558.60009
[Gui90] Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Theory Dyn. Syst., Volume 10 (1990) no. 3, pp. 483-512 | Article | MR 1074315
[Jon15] Dynamics of Berkovich spaces in low dimensions, Berkovich spaces and applications, Springer (Lecture Notes in Mathematics) Volume 2119 (2015), pp. 205-366 | Article | MR 3330767
[Kap09] Hyperbolic manifolds and discrete groups, Birkhäuser, Modern Birkhäuser Classics (2009), xxviii+467 pages (Reprint of the 2001 edition) | Article | MR 2553578
[LP89] Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 25 (1989) no. 2, pp. 109-142 | MR 1001021
[MT18] Random walks on weakly hyperbolic groups, J. Reine Angew. Math., Volume 742 (2018), pp. 187-239 | Article | MR 3849626
[Ota15] Compactification of spaces of representations after Culler, Morgan and Shalen, Berkovich spaces and applications, Springer (Lecture Notes in Mathematics) Volume 2119 (2015), pp. 367-413 | Article | MR 3330768
[Poi13] Les espaces de Berkovich sont angéliques, Bull. Soc. Math. Fr., Volume 141 (2013) no. 2, pp. 267-297 | Article | MR 3081557
[Tem15] Introduction to Berkovich analytic spaces, Berkovich spaces and applications, Springer (Lecture Notes in Mathematics) Volume 2119 (2015), pp. 3-66 | Article | MR 3330762
[YW15] On p-adic Möbius maps (2015) (https://arxiv.org/abs/1512.01305)