# zbMATH — the first resource for mathematics

Analytic reparametrization of semi-algebraic sets. (English) Zbl 1143.32007
The paper concerns the controlled parametrizations of compact semi-algebraic sets. The main result is a generalization, to the analytic case, of the following theorem (Yomdin, Gromov). For any natural $$k$$ and any compact semi-algebraic set $$A\subset I^{n}\subset \mathbb{R}^{n}$$ ($$I:=[-1,1]$$) there exists a subdivision of $$A$$ into semi-algebraic pieces $$A_{j},$$ $$j=1,\dots ,N,$$ together with algebraic $$C^{k}$$-mappings $$\psi _{j}:I^{n_{j}}\rightarrow A_{j}$$ such that $$\psi _{j}$$ are onto, homeomorphic on the interior of $$I^{n_{j}}$$ and $$\psi _{j}(x)-\psi _{j}(0)$$ are uniformly bounded by $$1$$ with the all derivatives up to the order $$k.$$ Moreover $$N$$ depends only on $$k$$ and the degrees and the number of equations and inequalities defining $$A.$$ The analytic case differs in that the $$\psi _{j}$$ are real analytic, uniformly bounded with the all derivatives but after removing from $$A$$ a finite number $$M$$ of open boxes of size at most $$2\delta$$ ($$\delta$$ is a given positive real number). Moreover $$M$$ and $$N$$ depends only on $$k$$ and the degrees and the number of equations and inequalities defining $$A$$ times the factor $$\log _{2}(\frac{1}{\delta }).$$

##### MSC:
 32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions 14P10 Semialgebraic sets and related spaces
Full Text:
##### References:
 [1] Afraimovich, V.; Glebsky, L., Measures of $$\varepsilon$$-complexity, Taiwanese J. math., 9, 3, 397-409, (2005) · Zbl 1094.28008 [2] Afraimovich, V.; Glebsky, L., Complexity, fractional dimension and topological entropy in dynamical systems, (), 35-72 [3] Baran, M.; Pleśniak, W., Bernstein and Van der Corput-shaake type inequalities on semialgebraic curves, Stud. math., 125, 1, 83-96, (1997) · Zbl 0895.41011 [4] Baran, M.; Pleśniak, M.W., Polynomial inequalities on algebraic sets, Stud. math., 141, 3, 209-219, (2000) · Zbl 0987.41006 [5] Baran, M.; Pleśniak, W., Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities, Stud. math., 141, 3, 221-234, (2000) · Zbl 0987.41005 [6] Benedetti, R.; Risler, J.J., Real algebraic and semi-algebraic sets, actualites mathematiques, (1990), Hermann Paris [7] Bernstein, S., Sur une propriété des polynomes, Proc. kharkov math. soc. ser. 2, 14, 1-6, (1913) [8] Bos, L.; Levenberg, N.; Milman, P.; Taylor, B.A., Tangential Markov inequalities characterize algebraic submanifolds of $$C^n$$, Indiana univ. math. J., 44, 115-137, (1995) · Zbl 0824.41015 [9] Bos, L.; Levenberg, N.; Milman, P.; Taylor, B.A., Tangential Markov inequalities on real algebraic varieties, Indiana univ. math. J., 47, 4, 1257-1272, (1998) · Zbl 0938.32008 [10] Bourgain, J., Estimates on Green’s functions, localization and the quantum kicked rotor model, Ann. math., 156, 249-294, (2002) · Zbl 1213.82054 [11] Bourgain, J.; Goldstein, M.; Schlag, W., Anderson localization for schrodinger operators on $$Z^2$$ with quasi-periodic potential, Acta math., 188, 41-86, (2002) · Zbl 1022.47023 [12] Briskin, M.; Yomdin, Y., Algebraic families of analytic functions, I. J. differ. equations, 136, 2, 248-267, (1997) · Zbl 0886.34005 [13] Browning, T.; Heath-Brown, D.; Salberger, P., Counting rational points on algebraic varieties, Duke math. J., 132, 3, 545-578, (2006), (11G35, 14G05) · Zbl 1098.14013 [14] Brudnyi, A., Bernstein-type inequality for algebraic functions, Indiana univ. math. J., 46, 1, 95-117, (1997) [15] D. Burguet, A proof of Gromov’s algebraic lemma, 2005, preprint. · Zbl 1169.14038 [16] Cowieson, W.; Young, L.-S., SRB measures as zero-noise limits, Ergodic theory dyn. syst., 25, 4, 1115-1138, (2005) · Zbl 1098.37020 [17] Dinh, T.-C.; Sibony, N., Dinamique des applications d’allure polynomiale, J. math. pure appl., 82, 367-423, (2003) · Zbl 1033.37023 [18] Dinh, T.-C.; Sibony, N., Une borne suprieure pour l’entropie topologique d’une application rationnelle, Ann. math. (2), 161, 3, 1637-1644, (2005) · Zbl 1084.54013 [19] Dinh, T.-C.; Sibony, N., Geometry of currents, intersection theory and dynamics of horizontal-like maps, Ann. inst. Fourier (Grenoble), 56, 2, 423-457, (2006) · Zbl 1089.37036 [20] Dinh, T.-C.; Sibony, N., Decay of correlations and the central limit theorem for meromorphic maps, Comm. pure appl. math., 59, 5, 754-768, (2006) · Zbl 1137.37023 [21] Downarowicz, T., Entropy structure, J. anal. math., 96, 57-116, (2005) · Zbl 1151.37020 [22] Downarowicz, T.; Newhouse, S., Symbolic extensions and smooth dynamical systems, Invent. math., 160, 3, 453-499, (2005) · Zbl 1067.37018 [23] Fefferman, C.; Narasimhan, R., Bernstein’s inequality on algebraic curves, Ann. inst. Fourier (Grenoble), 43, 5, 1319-1348, (1993) · Zbl 0842.26013 [24] Fefferman, C.; Narasimhan, R., On the polynomial-like behaviour of certain algebraic functions, Ann. inst. Fourier (Grenoble), 44, 2, 1091-1179, (1994) · Zbl 0811.14046 [25] Fefferman, C.; Narasimhan, R., Bernstein’s inequality and the resolution of spaces of analytic functions, Duke math. J., 81, 1, 77-98, (1995) · Zbl 0854.32006 [26] Fontich, E.; de la Llave, R.; Martn, P., Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems, Trans. am. math. soc., 358, 3, 1317-1345, (2006), (electronic) · Zbl 1080.37022 [27] Françoise, J.-P.; Yomdin, Y., Bernstein inequality and applications to differential equations and analytic geometry, J. funct. anal., 146, 1, 185-205, (1997) · Zbl 0869.34008 [28] Françoise, J.-P.; Yomdin, Y., Projection of analytic sets and Bernstein inequalities, (), 103-108 · Zbl 0915.30002 [29] Friedland, S., Entropy of polynomial and rational maps, Ann. math., 133, 359-368, (1991) · Zbl 0737.54006 [30] S. Friedland, Entropy of polynomial and rational maps: a survey, Arxiv, 2006. [31] Gromov, M., Spectral geometry of semi-algebraic sets, Ann. inst. Fourier (Grenoble), 42, 1-2, 249-274, (1992) · Zbl 0759.58048 [32] Gromov, M., On the entropy of holomorphic maps, Enseign. math (2), 49, 3-4, 217-235, (2003) · Zbl 1080.37051 [33] Gromov, M., Three remarks on geodesic dynamics and fundamental group, Preprint SUNY (1976). reprinted in $$\operatorname{L}$$’ enseighement mathematique, 46, 391-402, (2000) · Zbl 1002.53028 [34] M. Gromov, Entropy, homology and semialgebraic geometry (after Y. Yomdin), Sémin. Bourbaki, 1985/86. · Zbl 0611.58041 [35] Guedj, V., Entropie topologique des applications méromorphes, Ergodic theory dyn. syst., 25, 1847-1855, (2005) · Zbl 1087.37015 [36] Guedj, V.; Sibony, N., Dynamics of polynomial automorphisms of $$\mathbb{C}^k$$, Ark. mat., 40, 2, 207-243, (2002) · Zbl 1034.37025 [37] Hardt, R., Triangulation of subanalytic sets and proper light subanalytic maps, Invent. math., 38, 207-217, (1977) · Zbl 0331.32006 [38] H. Hironaka, Triangulations of algebraic sets, in: Proceedings of the Symposia in Pure Mathematics American Mathematical Society, vol. 29, Providence, RI, 1975, pp. 165-185. · Zbl 0332.14001 [39] Kruglikov, B.; Matveev, V., Strictly non-proportional geodesically equivalent metrics have $$h_{\operatorname{top}}(g) = 0$$, Ergodic theory dyn. syst., 26, 1, 247-266, (2006) · Zbl 1094.53037 [40] Kurdyka, K., On a subanalytic stratification satisfying a Whitney property with exponent 1. real algebraic geometry (Rennes, 1991), (), 316-322 [41] Kurdyka, K.; Mostowski, T.; Parusin‘ski, A., Proof of the gradient conjecture of R. thom, Ann. math. (2), 152, 3, 763-792, (2000) · Zbl 1053.37008 [42] Lord, G.J.; Rougemont, J., Topological and $$\varepsilon$$-entropy for large volume limits of discretized parabolic equations, SIAM J. numer. anal., 40, 4, 1311-1329, (2002) · Zbl 1024.35019 [43] McMullen, Dynamics on $$K 3$$-surfaces: salem numbers and Siegel disks, J. reine angew. math., 545, 201-233, (2002) · Zbl 1054.37026 [44] J. Milnor, Is entropy effectively computable?, in a site “Open Problems in Dynamics and Ergodic Theory”, http://iml.univ-mrs.fr/ kolyada/opds/. [45] Milnor, J.; Tresser, Ch., On entropy and monotonicity for real cubic maps, Comment. math. phys., 209, 123-278, (2000) · Zbl 0971.37007 [46] Paternain, G.; Petean, J., Zero entropy and bounded topology, Comment. math. helv., 81, 2, 287-304, (2006) · Zbl 1093.53090 [47] Pila, J., Integer points on the dilation of a subanalytic surface, Q. J. math., 55, Part 2, 207-223, (2004) · Zbl 1111.32004 [48] Pila, J., Rational points on a subanalytic surface, Ann. inst. Fourier (Grenoble), 55, 5, 1501-1516, (2005) · Zbl 1121.11032 [49] Pila, J.; Wilkie, A.J., The rational points of a definable set, Duke math. J., 133, 3, 591-616, (2006) · Zbl 1217.11066 [50] Roytvarf, N.; Yomdin, Y., Bernstein classes, Ann. inst. Fourier (Grenoble), 47, 3, 825-858, (1997) · Zbl 0974.30524 [51] N. Sibony, Dinamique des applications rationnelles de $$\mathbb{P}^k$$, Dinamique et géométrie complexes (Lyon, 1997), Panor, Syntheses, vol. 8, SMF, Paris, 1999, pp. 97-185. [52] de Thelin, H., Sur la construction de mesures selles, Ann. inst. Fourier (Grenoble), 56, 2, 337-372, (2006) · Zbl 1100.37029 [53] A.J. Wilkie, private communication. [54] A.J. Wilkie, Lecture notes, Cambridge, March-April 2005, $$\langle$$http://www.newton.cam.ac.uk/webseminars/pg+ws/2005/maa/0203/wilkie⟩. [55] Yomdin, Y., Volume growth and entropy, Israel J. math., 57, 3, 285-300, (1987) · Zbl 0641.54036 [56] Yomdin, Y., $$C^k$$-resolution of semialgebraic sets and mappings, Israel J. math., 57, 3, 301-317, (1987) · Zbl 0641.54037 [57] Yomdin, Y., Local complexity growth for iterations of real analytic mappings and semi-continuity moduli of the entropy, Ergodic theory dyn. syst., 11, 583-602, (1991) · Zbl 0756.58041 [58] Yomdin, Y., Complexity of functions: some questions, conjectures, and results, J. complexity, 7, 1, 70-96, (1991) · Zbl 0759.41022 [59] Yomdin, Y., Global finiteness properties of analytic families and algebra of their Taylor coefficients, (), 527-555 · Zbl 0944.34003 [60] Yomdin, Y., Semialgebraic complexity of functions, J. complexity, 21, 111-148, (2005) · Zbl 1101.68614 [61] D. Haviv, Y. Yomdin, Uniform approximation of near-singular surfaces, in the Proceedings of A. Galligo Conference, Nice, 2006, to appear. · Zbl 1134.68061 [62] Y. Yomdin, Analytic reparametrization of semi-algebraic sets and mappings, in preparation. · Zbl 1143.32007 [63] Y. Yomdin, Local complexity bounds in analytic dynamics, in preparation. · Zbl 0756.58041 [64] Zaslavsky, G.; Afraimovich, V., Working with complexity functions, (), 73-85 · Zbl 1136.37312
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.