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Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data
Diffeomorphisms with a generalized Lipschitz shadowing property
1. | Department of Mathematics, Mokwon University, Daejeon 35349, Korea |
2. | Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea |
3. | School of Mathematical Sciences, Beihang University, Beijing 100191, China |
Shadowing property and structural stability are important dynamics with close relationship. Pilyugin and Tikhomirov proved that Lipschitz shadowing property implies the structural stability[
References:
[1] |
S-T. Liao,
Obstruction sets I, Acta Math. Sinica., 23 (1980), 411-453.
|
[2] |
R. Mañé, Characterizations of AS diffeomorphisms, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Springer, Berlin, 597 (1977), 389–394. |
[3] |
R. Mañé,
A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161-210.
|
[4] |
S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706. Springer-Verlag, Berlin, 1999. |
[5] |
S. Y. Pilyugin and S. Tikhomirov,
Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515.
doi: 10.1088/0951-7715/23/10/009. |
[6] |
S. Y. Pilyugin and K. Sakai, Shadowing and Hyperbolicity, Lecture Notes in Math., 2193. Springer, Cham, 2017.
doi: 10.1007/978-3-319-65184-2. |
[7] |
V. A. Pliss, Bounded solutions of nonhomogeneous linear systems of differential equations, Problems in the Asymptotic Theory of Nonlinear Oscillations [in Russian], Naukova Dumka, Kiev, 278 (1977), 168–173. |
[8] |
C. Robinson,
Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 7 (1977), 425-437.
doi: 10.1216/RMJ-1977-7-3-425. |
[9] |
K. Sakai,
Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.
|
[10] |
D. Todorov,
Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory, Discrete Contin. Dyn. Syst., 33 (2013), 4187-4205.
doi: 10.3934/dcds.2013.33.4187. |
[11] |
L. Wen, Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity, Graduate Studies in Mathematics 173, American Mathematical Society, Providence, RI, 2016.
doi: 10.1090/gsm/173. |
[12] |
X. Wen, S. Gan and L. Wen,
$C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations., 246 (2009), 340-357.
doi: 10.1016/j.jde.2008.03.032. |
show all references
References:
[1] |
S-T. Liao,
Obstruction sets I, Acta Math. Sinica., 23 (1980), 411-453.
|
[2] |
R. Mañé, Characterizations of AS diffeomorphisms, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Springer, Berlin, 597 (1977), 389–394. |
[3] |
R. Mañé,
A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161-210.
|
[4] |
S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706. Springer-Verlag, Berlin, 1999. |
[5] |
S. Y. Pilyugin and S. Tikhomirov,
Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515.
doi: 10.1088/0951-7715/23/10/009. |
[6] |
S. Y. Pilyugin and K. Sakai, Shadowing and Hyperbolicity, Lecture Notes in Math., 2193. Springer, Cham, 2017.
doi: 10.1007/978-3-319-65184-2. |
[7] |
V. A. Pliss, Bounded solutions of nonhomogeneous linear systems of differential equations, Problems in the Asymptotic Theory of Nonlinear Oscillations [in Russian], Naukova Dumka, Kiev, 278 (1977), 168–173. |
[8] |
C. Robinson,
Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 7 (1977), 425-437.
doi: 10.1216/RMJ-1977-7-3-425. |
[9] |
K. Sakai,
Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.
|
[10] |
D. Todorov,
Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory, Discrete Contin. Dyn. Syst., 33 (2013), 4187-4205.
doi: 10.3934/dcds.2013.33.4187. |
[11] |
L. Wen, Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity, Graduate Studies in Mathematics 173, American Mathematical Society, Providence, RI, 2016.
doi: 10.1090/gsm/173. |
[12] |
X. Wen, S. Gan and L. Wen,
$C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations., 246 (2009), 340-357.
doi: 10.1016/j.jde.2008.03.032. |
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