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On the Bidomain equations driven by stochastic forces
Gromov-Hausdorff distances for dynamical systems
Department of Mathematics, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon-si, Gyeonggi-do, 16419, Korea |
We study equivariant Gromov-Hausdorff distances for general actions which are not necessarily isometric as Fukaya introduced. We prove that if an action is expansive and has the pseudo-orbit tracing property then it is stable under our adapted equivariant Gromov-Hausdorff topology. Finally, using Lott and Villani's ideas of optimal transport, we investigate equivariant Gromov-Hausdorff convergence for actions of locally compact amenable groups on Wasserstein spaces.
References:
[1] |
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Mathematical Library, 52, Recent advances, North-Holland Publishing Co., Amsterdam, 1994. |
[2] |
A. Arbieto and C. A. Morales Rojas,
Topological stability from Gromov-Hausdorff viewpoint, Discrete Contin. Dyn. Syst., 37 (2017), 3531-3544.
doi: 10.3934/dcds.2017151. |
[3] |
R. Bowen, Equilibrium States and The Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. |
[4] |
R. Bowen,
$\omega $-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
[5] |
R. Bowen and P. Walters,
Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.
doi: 10.1016/0022-0396(72)90013-7. |
[6] |
D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/033. |
[7] |
J. Cheeger and T. H. Colding,
Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), 144 (1996), 189-237.
doi: 10.2307/2118589. |
[8] |
J. Cheeger, K. Fukaya and M. Gromov,
Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc., 5 (1992), 327-372.
doi: 10.1090/S0894-0347-1992-1126118-X. |
[9] |
N.-P. Chung and K. Lee,
Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057.
doi: 10.1090/proc/13654. |
[10] |
N.-P. Chung and H. Li,
Homoclinic groups, IE groups, and expansive algebraic actions, Invent. Math., 199 (2015), 805-858.
doi: 10.1007/s00222-014-0524-1. |
[11] |
M. Dong, K. Lee and C. Morales, Gromov-Hausdorff perturbations of group actions, preprint. |
[12] |
M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 259, 2011.
doi: 10.1007/978-0-85729-021-2. |
[13] |
K. Fukaya,
Theory of convergence for Riemannian orbifolds, Japan. J. Math. (N.S.), 12 (1986), 121-160.
doi: 10.4099/math1924.12.121. |
[14] |
K. Fukaya,
A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom., 28 (1988), 1-21.
doi: 10.4310/jdg/1214442157. |
[15] |
K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math., Academic Press, Boston, MA, 18 (1990), 143–238.
doi: 10.2969/aspm/01810143. |
[16] |
K. Fukaya and T. Yamaguchi,
The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2), 136 (1992), 253-333.
doi: 10.2307/2946606. |
[17] |
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73. |
[18] |
K. Grove and P. Petersen, Manifolds near the boundary of existence, J. Differential Geom., 33 1991,379–394.
doi: 10.4310/jdg/1214446323. |
[19] |
J. Harvey,
Equivariant Alexandrov geometry and orbifold finiteness,, J. Geom. Anal., 26 (2016), 1925-1945.
doi: 10.1007/s12220-015-9614-6. |
[20] |
J. Harvey and C. Searle,
Orientation and symmetries of Alexandrov spaces with applications in positive curvature, J. Geom. Anal., 27 (2017), 1636-1666.
doi: 10.1007/s12220-016-9734-7. |
[21] |
A. G. Khan, P. Das and T. Das, GH-stability and spectral decomposition for group actions, arXiv: 1804.05920v3. |
[22] |
J. Lott and C. Villani,
Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991.
doi: 10.4007/annals.2009.169.903. |
[23] |
T. Meyerovitch,
Pseudo-orbit tracing and algebraic actions of countable amenable groups, Ergodic Theory Dynam. Systems, 39 (2019), 2570-2591.
doi: 10.1017/etds.2017.126. |
[24] |
P. Oprocha,
Shadowing in multi-dimensional shift spaces, Colloq. Math., 110 (2008), 451-460.
doi: 10.4064/cm110-2-8. |
[25] |
A. V. Osipov and S. B. Tikhomirov,
Shadowing for actions of some finitely generated groups, Dyn. Syst., 29 (2014), 337-351.
|
[26] |
A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, American Mathematical Society, Providence, RI, 1988.
doi: 10.1090/surv/029. |
[27] |
P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, 3, Springer, Cham, 2016.
doi: 10.1007/978-3-319-26654-1. |
[28] |
J.-P. Pier, Amenable Locally Compact Groups, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. |
[29] |
S. Yu. Pilyugin and S. B. Tikhomirov,
Shadowing in actions of some abelian groups, Fund. Math., 179 (2003), 83-96.
doi: 10.4064/fm179-1-7. |
[30] |
X. Rong, Convergence and Collapsing Theorems in Riemannian Geometry, Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM), 13, Int. Press, omerville, MA, 2010.
![]() ![]() |
[31] |
T. Shioya, Metric Measure Geometry, IRMA Lectures in Mathematics and Theoretical Physics, 25, Gromov's theory of convergence and concentration of metrics and measures, EMS Publishing House, Zürich, 2016.
doi: 10.4171/158. |
[32] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058. |
[33] |
C. Villani, Optimal Transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Old and new, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
[34] |
P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, Berlin, 668 (1978), 231–244. |
[35] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
show all references
References:
[1] |
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Mathematical Library, 52, Recent advances, North-Holland Publishing Co., Amsterdam, 1994. |
[2] |
A. Arbieto and C. A. Morales Rojas,
Topological stability from Gromov-Hausdorff viewpoint, Discrete Contin. Dyn. Syst., 37 (2017), 3531-3544.
doi: 10.3934/dcds.2017151. |
[3] |
R. Bowen, Equilibrium States and The Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. |
[4] |
R. Bowen,
$\omega $-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
[5] |
R. Bowen and P. Walters,
Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.
doi: 10.1016/0022-0396(72)90013-7. |
[6] |
D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/033. |
[7] |
J. Cheeger and T. H. Colding,
Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), 144 (1996), 189-237.
doi: 10.2307/2118589. |
[8] |
J. Cheeger, K. Fukaya and M. Gromov,
Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc., 5 (1992), 327-372.
doi: 10.1090/S0894-0347-1992-1126118-X. |
[9] |
N.-P. Chung and K. Lee,
Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057.
doi: 10.1090/proc/13654. |
[10] |
N.-P. Chung and H. Li,
Homoclinic groups, IE groups, and expansive algebraic actions, Invent. Math., 199 (2015), 805-858.
doi: 10.1007/s00222-014-0524-1. |
[11] |
M. Dong, K. Lee and C. Morales, Gromov-Hausdorff perturbations of group actions, preprint. |
[12] |
M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 259, 2011.
doi: 10.1007/978-0-85729-021-2. |
[13] |
K. Fukaya,
Theory of convergence for Riemannian orbifolds, Japan. J. Math. (N.S.), 12 (1986), 121-160.
doi: 10.4099/math1924.12.121. |
[14] |
K. Fukaya,
A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom., 28 (1988), 1-21.
doi: 10.4310/jdg/1214442157. |
[15] |
K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math., Academic Press, Boston, MA, 18 (1990), 143–238.
doi: 10.2969/aspm/01810143. |
[16] |
K. Fukaya and T. Yamaguchi,
The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2), 136 (1992), 253-333.
doi: 10.2307/2946606. |
[17] |
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73. |
[18] |
K. Grove and P. Petersen, Manifolds near the boundary of existence, J. Differential Geom., 33 1991,379–394.
doi: 10.4310/jdg/1214446323. |
[19] |
J. Harvey,
Equivariant Alexandrov geometry and orbifold finiteness,, J. Geom. Anal., 26 (2016), 1925-1945.
doi: 10.1007/s12220-015-9614-6. |
[20] |
J. Harvey and C. Searle,
Orientation and symmetries of Alexandrov spaces with applications in positive curvature, J. Geom. Anal., 27 (2017), 1636-1666.
doi: 10.1007/s12220-016-9734-7. |
[21] |
A. G. Khan, P. Das and T. Das, GH-stability and spectral decomposition for group actions, arXiv: 1804.05920v3. |
[22] |
J. Lott and C. Villani,
Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991.
doi: 10.4007/annals.2009.169.903. |
[23] |
T. Meyerovitch,
Pseudo-orbit tracing and algebraic actions of countable amenable groups, Ergodic Theory Dynam. Systems, 39 (2019), 2570-2591.
doi: 10.1017/etds.2017.126. |
[24] |
P. Oprocha,
Shadowing in multi-dimensional shift spaces, Colloq. Math., 110 (2008), 451-460.
doi: 10.4064/cm110-2-8. |
[25] |
A. V. Osipov and S. B. Tikhomirov,
Shadowing for actions of some finitely generated groups, Dyn. Syst., 29 (2014), 337-351.
|
[26] |
A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, American Mathematical Society, Providence, RI, 1988.
doi: 10.1090/surv/029. |
[27] |
P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, 3, Springer, Cham, 2016.
doi: 10.1007/978-3-319-26654-1. |
[28] |
J.-P. Pier, Amenable Locally Compact Groups, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. |
[29] |
S. Yu. Pilyugin and S. B. Tikhomirov,
Shadowing in actions of some abelian groups, Fund. Math., 179 (2003), 83-96.
doi: 10.4064/fm179-1-7. |
[30] |
X. Rong, Convergence and Collapsing Theorems in Riemannian Geometry, Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM), 13, Int. Press, omerville, MA, 2010.
![]() ![]() |
[31] |
T. Shioya, Metric Measure Geometry, IRMA Lectures in Mathematics and Theoretical Physics, 25, Gromov's theory of convergence and concentration of metrics and measures, EMS Publishing House, Zürich, 2016.
doi: 10.4171/158. |
[32] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058. |
[33] |
C. Villani, Optimal Transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Old and new, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
[34] |
P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, Berlin, 668 (1978), 231–244. |
[35] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
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