January  2002, 8(1): 17-28. doi: 10.3934/dcds.2002.8.17

Global inversion via the Palais-Smale condition

1. 

Department of Mathematics, Texas Christian University, Fort Worth, TX 76129, United States

2. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States

Received  March 2001 Revised  October 2001 Published  October 2001

Fixing a complete Riemannian metric g on $\mathbb R^n$, we show that a local diffeomorphism $f : \mathbb R^n\to \mathbb R^n$ is bijective if the height function $f\cdot v$ (standard inner product) satisfies the Palais-Smale condition relative to $g$ for each for each nonzero $v\in \mathbb R^n$. Our method substantially improves a global inverse function theorem of Hadamard. In the context of polynomial maps, we obtain new criteria for invertibility in terms of Lojasiewicz exponents and tameness of polynomials.
Citation: Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17
[1]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829

[2]

A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987

[3]

Laura Poggiolini, Marco Spadini. Local inversion of a class of piecewise regular maps. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2207-2224. doi: 10.3934/cpaa.2018105

[4]

Ronen Peretz, Nguyen Van Chau, L. Andrew Campbell, Carlos Gutierrez. Iterated images and the plane Jacobian conjecture. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 455-461. doi: 10.3934/dcds.2006.16.455

[5]

François Béguin. Smale diffeomorphisms of surfaces: a classification algorithm. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 261-310. doi: 10.3934/dcds.2004.11.261

[6]

Neil S. Trudinger. On the local theory of prescribed Jacobian equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1663-1681. doi: 10.3934/dcds.2014.34.1663

[7]

Zsolt Páles, Vera Zeidan. $V$-Jacobian and $V$-co-Jacobian for Lipschitzian maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 623-646. doi: 10.3934/dcds.2011.29.623

[8]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[9]

Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489

[10]

Xu Zhang, Guanrong Chen. Polynomial maps with hidden complex dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2941-2954. doi: 10.3934/dcdsb.2018293

[11]

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

[12]

Francisco Braun, José Ruidival dos Santos Filho. The real jacobian conjecture on $\R^2$ is true when one of the components has degree 3. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 75-87. doi: 10.3934/dcds.2010.26.75

[13]

Eric Bedford, Kyounghee Kim. Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 977-1013. doi: 10.3934/dcds.2008.21.977

[14]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[15]

Huaibin Li. An equivalent characterization of the summability condition for rational maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4567-4578. doi: 10.3934/dcds.2013.33.4567

[16]

Shenggui Zhang. A sufficient condition of Euclidean rings given by polynomial optimization over a box. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 93-101. doi: 10.3934/naco.2014.4.93

[17]

Krerley Oliveira, Marcelo Viana. Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 225-236. doi: 10.3934/dcds.2006.15.225

[18]

Can Gao, Joachim Krieger. Optimal polynomial blow up range for critical wave maps. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1705-1741. doi: 10.3934/cpaa.2015.14.1705

[19]

Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793

[20]

Mary Wilkerson. Thurston's algorithm and rational maps from quadratic polynomial matings. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2403-2433. doi: 10.3934/dcdss.2019151

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]