-
Previous Article
A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion
- DCDS Home
- This Issue
-
Next Article
Large data global regularity for the classical equivariant Skyrme model
Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry
Institut für Algebra und Geometrie, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76 131 Karlsruhe, Germany |
Let $X$ be a proper Hadamard space and $\Gamma <{\text{Is}}(X)$ a non-elementary discrete group of isometries with a rank one isometry. We discuss and prove Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on the set of parametrized geodesics of the quotient $\Gamma \backslash X$ and with respect to Ricks' measure introduced in [
References:
[1] |
J. Aaronson and M. Denker,
The Poincaré series of $\mathbb C\setminus \mathbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20.
doi: 10.1017/S0143385799126592. |
[2] |
J. Aaronson and D. Sullivan,
Rational ergodicity of geodesic flows, Ergodic Theory Dynam. Systems, 4 (1984), 165-178.
doi: 10.1017/S0143385700002364. |
[3] |
W. Ballmann,
Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.
doi: 10.1007/BF01456836. |
[4] |
W. Ballmann,
Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609.
doi: 10.2307/1971331. |
[5] |
W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin.
doi: 10.1007/978-3-0348-9240-7. |
[6] |
W. Ballmann and M. Brin,
Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math., 82 (1995), 169-209.
|
[7] |
W. Ballmann, M. Brin and P. Eberlein,
Structure of manifolds of nonpositive curvature. Ⅰ, Ann. of Math. (2), 122 (1985), 171-203.
doi: 10.2307/1971373. |
[8] |
W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, vol. 61 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1985.
doi: 10.1007/978-1-4684-9159-3. |
[9] |
V. Bangert and V. Schroeder,
Existence of flat tori in analytic manifolds of nonpositive curvature, Ann. Sci. École Norm. Sup. (4), 24 (1991), 605-634.
doi: 10.24033/asens.1638. |
[10] |
M. Bourdon,
Structure conforme au bord et flot géodésique d'un $\rm CAT(-1)$-espace, Enseign. Math. (2), 41 (1995), 63-102.
|
[11] |
M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999.
doi: 10.1007/978-3-662-12494-9. |
[12] |
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. |
[13] |
K. Burns,
Hyperbolic behaviour of geodesic flows on manifolds with no focal points, Ergodic Theory Dynam. Systems, 3 (1983), 1-12.
doi: 10.1017/S0143385700001796. |
[14] |
K. Burns and R. Spatzier,
Manifolds of nonpositive curvature and their buildings, Inst. Hautes Etudes Sci. Publ. Math., 65 (1987), 35-59.
|
[15] |
P-E. Caprace and K. Fujiwara,
Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal., 19 (2010), 1296-1319.
doi: 10.1007/s00039-009-0042-2. |
[16] |
M. Coornaert and A. Papadopoulos,
Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d'isométries des arbres, Trans. Amer. Math. Soc., 343 (1994), 883-898.
doi: 10.2307/2154747. |
[17] |
F. Dal'bo, J.-P. Otal and M. Peigné,
Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124.
doi: 10.1007/BF02803518. |
[18] |
U. Hamenstädt, Rank-one isometries of proper CAT(0)-spaces, Discrete Groups and Geometric Structures, Contemporary Mathematics,, American Mathematical Society, Providence, RI, 501 (2009), 43–59.
doi: 10.1090/conm/501/09839. |
[19] |
E. Hopf, Ergodentheorie, Springer, 1937.
doi: 10.1007/978-3-642-86630-2. |
[20] |
E. Hopf,
Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc., 77 (1971), 863-877.
doi: 10.1090/S0002-9904-1971-12799-4. |
[21] |
V. A. Kaimanovich,
Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math., 455 (1994), 57-103.
doi: 10.1515/crll.1994.455.57. |
[22] |
G. Knieper,
On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.
doi: 10.1007/s000390050025. |
[23] |
G. Knieper,
The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds, Ann. of Math. (2), 148 (1998), 291-314.
doi: 10.2307/120995. |
[24] |
U. Krengel,
Darstellungssätze für Strömungen und Halbströmungen. Ⅰ, Mathematische Annalen, 176 (1968), 181-190.
doi: 10.1007/BF02052824. |
[25] |
U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1985, With a supplement by Antoine Brunel.
doi: 10.1515/9783110844641. |
[26] |
G. Link,
Asymptotic geometry and growth of conjugacy classes of nonpositively curved manifolds, Ann. Global Anal. Geom., 31 (2007), 37-57.
doi: 10.1007/s10455-006-9016-x. |
[27] |
G. Link,
Asymptotic geometry in products of Hadamard spaces with rank one isometries, Geometry and Topology, 14 (2010), 1063-1094.
doi: 10.2140/gt.2010.14.1063. |
[28] |
G. Link, M. Peigné and J.-C. Picaud,
Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006), 3-36.
|
[29] |
G. Link and J.-C. Picaud, Ergodic geometry for non-elementary rank one manifolds, Discrete and Continuous Dyn. Syst. A, no. 11, 36 (2016), 6257–6284.
doi: 10.3934/dcds.2016072. |
[30] |
P. J. Nicholls, The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511600678. |
[31] |
J.-P. Otal and M. Peigné,
Principe variationnel et groupes kleiniens, Duke Math. J., 125 (2004), 15-44.
doi: 10.1215/S0012-7094-04-12512-6. |
[32] |
S. J. Patterson,
The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.
doi: 10.1007/BF02392046. |
[33] |
M. Peigné, Autour de l'exposant de Poincaré d'un groupe kleinien, Géométrie ergodique, Monogr. Enseign. Math., Enseignement Math., Geneva, 43 (2013), 25–59. |
[34] |
R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one spaces, PhD Thesis, University of Michigan, 2015. Google Scholar |
[35] |
R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces, Ergodic Theory Dynam. Systems, no. 3, 37 (2017), 939–970.
doi: 10.1017/etds.2015.78. |
[36] |
T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N. S. ), 95 (2003), ⅵ+96pp. |
[37] |
T. Roblin,
Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math., 147 (2005), 333-357.
doi: 10.1007/BF02785371. |
[38] |
V. Schroeder,
Existence of immersed tori in manifolds of nonpositive curvature, J. Reine Angew. Math., 390 (1988), 32-46.
doi: 10.1515/crll.1988.390.32. |
[39] |
V. Schroeder,
Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989), 77-105.
doi: 10.1007/BF01182086. |
[40] |
V. Schroeder,
Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990), 251-263.
doi: 10.1007/BF00181332. |
[41] |
D. Sullivan,
The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.
|
[42] |
M. E. Taylor, Measure Theory and Integration, vol. 76 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/gsm/076. |
[43] |
M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original. |
[44] |
C. Yue,
The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005.
doi: 10.1090/S0002-9947-96-01614-5. |
show all references
References:
[1] |
J. Aaronson and M. Denker,
The Poincaré series of $\mathbb C\setminus \mathbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20.
doi: 10.1017/S0143385799126592. |
[2] |
J. Aaronson and D. Sullivan,
Rational ergodicity of geodesic flows, Ergodic Theory Dynam. Systems, 4 (1984), 165-178.
doi: 10.1017/S0143385700002364. |
[3] |
W. Ballmann,
Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.
doi: 10.1007/BF01456836. |
[4] |
W. Ballmann,
Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609.
doi: 10.2307/1971331. |
[5] |
W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin.
doi: 10.1007/978-3-0348-9240-7. |
[6] |
W. Ballmann and M. Brin,
Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math., 82 (1995), 169-209.
|
[7] |
W. Ballmann, M. Brin and P. Eberlein,
Structure of manifolds of nonpositive curvature. Ⅰ, Ann. of Math. (2), 122 (1985), 171-203.
doi: 10.2307/1971373. |
[8] |
W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, vol. 61 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1985.
doi: 10.1007/978-1-4684-9159-3. |
[9] |
V. Bangert and V. Schroeder,
Existence of flat tori in analytic manifolds of nonpositive curvature, Ann. Sci. École Norm. Sup. (4), 24 (1991), 605-634.
doi: 10.24033/asens.1638. |
[10] |
M. Bourdon,
Structure conforme au bord et flot géodésique d'un $\rm CAT(-1)$-espace, Enseign. Math. (2), 41 (1995), 63-102.
|
[11] |
M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999.
doi: 10.1007/978-3-662-12494-9. |
[12] |
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. |
[13] |
K. Burns,
Hyperbolic behaviour of geodesic flows on manifolds with no focal points, Ergodic Theory Dynam. Systems, 3 (1983), 1-12.
doi: 10.1017/S0143385700001796. |
[14] |
K. Burns and R. Spatzier,
Manifolds of nonpositive curvature and their buildings, Inst. Hautes Etudes Sci. Publ. Math., 65 (1987), 35-59.
|
[15] |
P-E. Caprace and K. Fujiwara,
Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal., 19 (2010), 1296-1319.
doi: 10.1007/s00039-009-0042-2. |
[16] |
M. Coornaert and A. Papadopoulos,
Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d'isométries des arbres, Trans. Amer. Math. Soc., 343 (1994), 883-898.
doi: 10.2307/2154747. |
[17] |
F. Dal'bo, J.-P. Otal and M. Peigné,
Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124.
doi: 10.1007/BF02803518. |
[18] |
U. Hamenstädt, Rank-one isometries of proper CAT(0)-spaces, Discrete Groups and Geometric Structures, Contemporary Mathematics,, American Mathematical Society, Providence, RI, 501 (2009), 43–59.
doi: 10.1090/conm/501/09839. |
[19] |
E. Hopf, Ergodentheorie, Springer, 1937.
doi: 10.1007/978-3-642-86630-2. |
[20] |
E. Hopf,
Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc., 77 (1971), 863-877.
doi: 10.1090/S0002-9904-1971-12799-4. |
[21] |
V. A. Kaimanovich,
Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math., 455 (1994), 57-103.
doi: 10.1515/crll.1994.455.57. |
[22] |
G. Knieper,
On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.
doi: 10.1007/s000390050025. |
[23] |
G. Knieper,
The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds, Ann. of Math. (2), 148 (1998), 291-314.
doi: 10.2307/120995. |
[24] |
U. Krengel,
Darstellungssätze für Strömungen und Halbströmungen. Ⅰ, Mathematische Annalen, 176 (1968), 181-190.
doi: 10.1007/BF02052824. |
[25] |
U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1985, With a supplement by Antoine Brunel.
doi: 10.1515/9783110844641. |
[26] |
G. Link,
Asymptotic geometry and growth of conjugacy classes of nonpositively curved manifolds, Ann. Global Anal. Geom., 31 (2007), 37-57.
doi: 10.1007/s10455-006-9016-x. |
[27] |
G. Link,
Asymptotic geometry in products of Hadamard spaces with rank one isometries, Geometry and Topology, 14 (2010), 1063-1094.
doi: 10.2140/gt.2010.14.1063. |
[28] |
G. Link, M. Peigné and J.-C. Picaud,
Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006), 3-36.
|
[29] |
G. Link and J.-C. Picaud, Ergodic geometry for non-elementary rank one manifolds, Discrete and Continuous Dyn. Syst. A, no. 11, 36 (2016), 6257–6284.
doi: 10.3934/dcds.2016072. |
[30] |
P. J. Nicholls, The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511600678. |
[31] |
J.-P. Otal and M. Peigné,
Principe variationnel et groupes kleiniens, Duke Math. J., 125 (2004), 15-44.
doi: 10.1215/S0012-7094-04-12512-6. |
[32] |
S. J. Patterson,
The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.
doi: 10.1007/BF02392046. |
[33] |
M. Peigné, Autour de l'exposant de Poincaré d'un groupe kleinien, Géométrie ergodique, Monogr. Enseign. Math., Enseignement Math., Geneva, 43 (2013), 25–59. |
[34] |
R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one spaces, PhD Thesis, University of Michigan, 2015. Google Scholar |
[35] |
R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces, Ergodic Theory Dynam. Systems, no. 3, 37 (2017), 939–970.
doi: 10.1017/etds.2015.78. |
[36] |
T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N. S. ), 95 (2003), ⅵ+96pp. |
[37] |
T. Roblin,
Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math., 147 (2005), 333-357.
doi: 10.1007/BF02785371. |
[38] |
V. Schroeder,
Existence of immersed tori in manifolds of nonpositive curvature, J. Reine Angew. Math., 390 (1988), 32-46.
doi: 10.1515/crll.1988.390.32. |
[39] |
V. Schroeder,
Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989), 77-105.
doi: 10.1007/BF01182086. |
[40] |
V. Schroeder,
Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990), 251-263.
doi: 10.1007/BF00181332. |
[41] |
D. Sullivan,
The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.
|
[42] |
M. E. Taylor, Measure Theory and Integration, vol. 76 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/gsm/076. |
[43] |
M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original. |
[44] |
C. Yue,
The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005.
doi: 10.1090/S0002-9947-96-01614-5. |
[1] |
Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 |
[2] |
Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390 |
[3] |
Fei Liu, Xiaokai Liu, Fang Wang. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021057 |
[4] |
Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004 |
[5] |
Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 |
[6] |
Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 |
[7] |
Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226 |
[8] |
Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025 |
[9] |
Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027 |
[10] |
Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, 2021, 15 (3) : 519-537. doi: 10.3934/ipi.2021003 |
[11] |
Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 |
[12] |
Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001 |
[13] |
Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81 |
[14] |
Huaning Liu, Xi Liu. On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021008 |
[15] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[16] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[17] |
Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 |
[18] |
Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021031 |
[19] |
Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 |
[20] |
Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]