On a structure of the fixed point set of homogeneous maps

Previous Article
Unbounded sequences of cycles in general autonomous equations with periodic nonlinearities
DCDS-S Home
This Issue
Next Article
Pointwise estimates for solutions of singular quasi-linear parabolic equations

April 2013, 6(4): 1017-1027. doi: 10.3934/dcdss.2013.6.1017

On a structure of the fixed point set of homogeneous maps

Yakov Krasnov ^1,2,3, , Alexander Kononovich ^1,2,3, and Grigory Osharovich ^1,2,3,

1.	Department of Mathematics
2.	Bar-Ilan University
3.	Ramat-Gan, 52900

Received June 2011 Revised September 2011 Published December 2012

Abstract
Related Papers

A spectral and inverse spectral problem for homogeneous polynomial maps is discussed.The $m$-independence of vectors based on the symmetric tensor powers performs as a main toolto study the structure of the spectrum. Possible restrictions on this structureare described in terms of syzygies provided by the Euler-Jacobi formula.Applications to projective dynamics are discussed.

Keywords: Fixed points theory, Bzout theorem, Euler-Jacobi formula, multi-linear analysis, syzygies..

Mathematics Subject Classification: Primary: 34A34, 15A69; Secondary: 15A18, 34C14, 34C4.

Citation: Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017

References:

[1]	B. Aupetit, Projections in real Banach algebras,, Bull. London Math. Soc., 13 (1981), 412. doi: 10.1112/blms/13.5.412. Google Scholar
[2]	M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes: II. Applications,, Ann. of Math. (2), 88 (1968), 451. Google Scholar
[3]	Z. Balanov and Y. Krasnov, Complex structures in real algebras I. Two-dimensional commutative case,, Comm. Algebra, 31 (2003), 4571. doi: 10.1081/AGB-120022810. Google Scholar
[4]	Z. Balanov, Y. Krasnov and A. Kononovich, Projective dynamics of homogeneous systems: Local invariants, syzygies and global residue theorem,, Z, 55 (2012), 577. Google Scholar
[5]	A. Dold, "Lectures on Algebraic Topology,", Berlin, (1974). Google Scholar
[6]	J. Esterle and J. Giol, Polynomial and polygonal connections between idempotents in finite dimensional real algebras,, Bull. London Math. Soc., 36 (2004), 378. doi: 10.1112/S0024609303002820. Google Scholar
[7]	W. Fulton, "Intersection Theory,", Second edition, 2 (1998). doi: 10.1007/978-1-4612-1700-8. Google Scholar
[8]	Z. V. Kovarik, Similarity and interpolation between projectors,, Acta Sci. Math. (Szeged), 39 (1977), 341. Google Scholar
[9]	J. Llibre and V. Pilyugina, Number of invariant straight Lines for homogeneous polynomial vector fields of arbitrary degree and dimension,, J. Dyn. Diff. Equat., 21 (2009), 487. doi: 10.1007/s10884-009-9141-x. Google Scholar
[10]	I. R. Shafarevich, "Basic Algebraic Geometry,", Berlin, 213 (1974). Google Scholar
[11]	M. Shub and S. Smale, Complexity of Bézout's theorem. I. Geometric aspects,, J. Amer. Math. Soc., 6 (1993), 459. doi: 10.2307/2152805. Google Scholar
[12]	A. Tretyakov and H. .Zołądek, A remark about homogeneous polynomial maps,, Topological Methods in Nonlinear Analysis, 19 (2002), 257. Google Scholar
[13]	H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math., 66 (1957), 545. Google Scholar
[14]	J. Zemánek, Idempotents in Banach algebras,, Bull. London Math. Soc., 11 (1979), 177. doi: 10.1112/blms/11.2.177. Google Scholar

show all references

References:

[1]	B. Aupetit, Projections in real Banach algebras,, Bull. London Math. Soc., 13 (1981), 412. doi: 10.1112/blms/13.5.412. Google Scholar
[2]	M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes: II. Applications,, Ann. of Math. (2), 88 (1968), 451. Google Scholar
[3]	Z. Balanov and Y. Krasnov, Complex structures in real algebras I. Two-dimensional commutative case,, Comm. Algebra, 31 (2003), 4571. doi: 10.1081/AGB-120022810. Google Scholar
[4]	Z. Balanov, Y. Krasnov and A. Kononovich, Projective dynamics of homogeneous systems: Local invariants, syzygies and global residue theorem,, Z, 55 (2012), 577. Google Scholar
[5]	A. Dold, "Lectures on Algebraic Topology,", Berlin, (1974). Google Scholar
[6]	J. Esterle and J. Giol, Polynomial and polygonal connections between idempotents in finite dimensional real algebras,, Bull. London Math. Soc., 36 (2004), 378. doi: 10.1112/S0024609303002820. Google Scholar
[7]	W. Fulton, "Intersection Theory,", Second edition, 2 (1998). doi: 10.1007/978-1-4612-1700-8. Google Scholar
[8]	Z. V. Kovarik, Similarity and interpolation between projectors,, Acta Sci. Math. (Szeged), 39 (1977), 341. Google Scholar
[9]	J. Llibre and V. Pilyugina, Number of invariant straight Lines for homogeneous polynomial vector fields of arbitrary degree and dimension,, J. Dyn. Diff. Equat., 21 (2009), 487. doi: 10.1007/s10884-009-9141-x. Google Scholar
[10]	I. R. Shafarevich, "Basic Algebraic Geometry,", Berlin, 213 (1974). Google Scholar
[11]	M. Shub and S. Smale, Complexity of Bézout's theorem. I. Geometric aspects,, J. Amer. Math. Soc., 6 (1993), 459. doi: 10.2307/2152805. Google Scholar
[12]	A. Tretyakov and H. .Zołądek, A remark about homogeneous polynomial maps,, Topological Methods in Nonlinear Analysis, 19 (2002), 257. Google Scholar
[13]	H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math., 66 (1957), 545. Google Scholar
[14]	J. Zemánek, Idempotents in Banach algebras,, Bull. London Math. Soc., 11 (1979), 177. doi: 10.1112/blms/11.2.177. Google Scholar

[1]	JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042
[2]	Paula Kemp. Fixed points and complete lattices. Conference Publications, 2007, 2007 (Special) : 568-572. doi: 10.3934/proc.2007.2007.568
[3]	John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443
[4]	Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557
[5]	Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761
[6]	Behrouz Kheirfam, Kamal mirnia. Multi-parametric sensitivity analysis in piecewise linear fractional programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 343-351. doi: 10.3934/jimo.2008.4.343
[7]	Jacek Banasiak, Aleksandra Puchalska. Generalized network transport and Euler-Hille formula. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1873-1893. doi: 10.3934/dcdsb.2018185
[8]	José Natário. An elementary derivation of the Montgomery phase formula for the Euler top. Journal of Geometric Mechanics, 2010, 2 (1) : 113-118. doi: 10.3934/jgm.2010.2.113
[9]	Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517
[10]	Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345
[11]	Francisco Crespo, Sebastián Ferrer. On the extended Euler system and the Jacobi and Weierstrass elliptic functions. Journal of Geometric Mechanics, 2015, 7 (2) : 151-168. doi: 10.3934/jgm.2015.7.151
[12]	David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205
[13]	Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[14]	Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121
[15]	Piotr Fijałkowski. A global inversion theorem for functions with singular points. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 173-180. doi: 10.3934/dcdsb.2018011
[16]	Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
[17]	Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583
[18]	Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
[19]	Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038
[20]	Marian Gidea, Yitzchak Shmalo. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6123-6148. doi: 10.3934/dcds.2018264

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences