doi: 10.3934/dcdss.2020074

A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces

Institute of Mathematics, NASU, Tereshchenkivs'ka 3, 01601 Kyiv, Ukraine

Dedicated to Professor Jürgen Scheurle on the occasion of his 65th birthday
Editors' note: Professor Sergiĭ Kolyada passed away on May 16, 2018. He will be missed by the mathematical community, as a mathematician and as a person. Due to his untimely death, Professor Kolyada could not implement the changes to the first version of the manuscript as suggested in the (positive) reviews. With the consent of Professor Kolyada's family, Professor L'ubomír Snoha (Matej Bel University, Banská Bystrica, Slovakia), a friend and colleague of Professor Kolyada, assumed the responsibility of carrying out the revision. The editors thank Professor Snoha for this invaluable contribution

Received  January 2018 Revised  September 2018 Published  April 2019

Fund Project: This survey is based on lectures given by the author at the Max Planck Institute for Mathematics, Technical University of Munich, Paris-Sud University, Luminy Institute of Mathematics, Institute of Mathematics of Jussieu and several other mathematical departments in 2017

The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is "Topological Dynamics". Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area has been recently called "Dynamical Topology". As an illustration, some topological properties of the space of all transitive interval maps are described. For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we survey some results on two new notions: "Slovak Space" and "Dynamical Compactness". A Slovak space, as a dynamical analogue of a rigid space, is a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in the theory of dynamical systems and means the collection of states which "attract" this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness is that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.

Citation: Sergiĭ Kolyada. A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020074
References:
[1] E. Akin, Recurrence in Topological Dynamics. Furstenberg families and Ellis actions, The University Series in Mathematics, Plenum Press, New York, 1997.
[2]

E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286. doi: 10.1007/BF02788112. Google Scholar

[3]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313. Google Scholar

[4]

E. Akin and J. Rautio, Chain transitive homeomorphisms on a space: All or none, Pacific J. Math., 291 (2017), 1-49. doi: 10.2140/pjm.2017.291.1. Google Scholar

[5]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988, Notas de Matemática [Mathematical Notes], 122. Google Scholar

[6]

J. AuslanderS. Kolyada and L'. Snoha, Functional envelope of a dynamical system, Nonlinearity, 20 (2007), 2245-2269. doi: 10.1088/0951-7715/20/9/012. Google Scholar

[7]

J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J., (2) 32 (1980), 177-188. doi: 10.2748/tmj/1178229634. Google Scholar

[8]

H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math., 60 (1967), 241-249. doi: 10.4064/fm-60-3-241-249. Google Scholar

[9]

T. DasE. Shah and L'. Snoha, (Non-)expansivity in functional envelopes, J. Math. Anal. Appl., 410 (2014), 1043-1048. doi: 10.1016/j.jmaa.2013.08.057. Google Scholar

[10]

T. Dobrowolski, Examples of topological groups homeomorphic to $l_2^f$, Proc. Amer. Math. Soc., 98 (1986), 303-311. doi: 10.2307/2045703. Google Scholar

[11]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc., (3) 4 (1954), 168-176. Google Scholar

[12]

T. Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and Topological Dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, 7-37. doi: 10.1090/conm/385/07188. Google Scholar

[13]

T. DownarowiczL'. Snoha and D. Tywoniuk, Minimal spaces with cyclic group of homeomorphisms, J. Dynam. Differential Equations, 29 (2017), 243-257. doi: 10.1007/s10884-015-9433-2. Google Scholar

[14]

F. T. Farrell and A. Gogolev, The space of Anosov diffeomorphisms, J. Lond. Math. Soc., (2) 89 (2014), 383-396. doi: 10.1112/jlms/jdt073. Google Scholar

[15]

A. Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup., 13 (1980), 45-93. doi: 10.24033/asens.1377. Google Scholar

[16]

B. R. Fayad, Topologically mixing and minimal but not ergodic, analytic transformation on ${{\rm{T}}^5}$, Bol. Soc. Brasil. Mat. (N.S.), 31 (2000), 277-285. doi: 10.1007/BF01241630. Google Scholar

[17]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49. doi: 10.1007/BF01692494. Google Scholar

[18] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.
[19]

H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math., 34 (1978), 61-85 (1979). doi: 10.1007/BF02790008. Google Scholar

[20]

P. Gartside and A. Glyn, Autohomeomorphism groups, Topology Appl., 129 (2003), 103-110. doi: 10.1016/S0166-8641(02)00140-2. Google Scholar

[21]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075. doi: 10.1088/0951-7715/6/6/014. Google Scholar

[22]

J. de Groot and R. J. Wille, Rigid continua and topological group-pictures, Arch. Math., 9 (1958), 441-446. doi: 10.1007/BF01898628. Google Scholar

[23]

J. de Groot, Groups represented by homeomorphism groups, Math. Ann., 138 (1959), 80-102. doi: 10.1007/BF01369667. Google Scholar

[24]

J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys., 70 (1979), 133-160. doi: 10.1007/BF01982351. Google Scholar

[25]

S. Harada, Remarks on the topological group of measure preserving transformations, Proc. Japan Acad., 27 (1951), 523-526. doi: 10.3792/pja/1195571228. Google Scholar

[26]

K. H. Hofmann and S. A. Morris, Compact homeomorphism groups are profinite, Topology Appl., 159 (2012), 2453-2462. doi: 10.1016/j.topol.2011.09.050. Google Scholar

[27]

K. H. Hofmann and S. A. Morris, Representing a profinite group as the homeomorphism group of a continuum, preprint, arXiv: 1108.3876.Google Scholar

[28]

W. HuangD. KhilkoS. Kolyada and G. Zhang, Dynamical compactness and sensitivity, J. Differential Equations, 260 (2016), 6800-6827. doi: 10.1016/j.jde.2016.01.011. Google Scholar

[29]

W. HuangD. KhilkoS. KolyadaA. Peris and G. Zhang, Finite intersection property and dynamical compactness, J. Dynam. Differential Equations, 30 (2018), 1221-1245. doi: 10.1007/s10884-017-9600-8. Google Scholar

[30]

W. HuangS. Kolyada and G. Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, Ergodic Theory Dynam. Systems, 38 (2018), 651-665. doi: 10.1017/etds.2016.48. Google Scholar

[31]

M. Keane, Contractibility of the automorphism group of a nonatomic measure space, Proc. Amer. Math. Soc., 26 (1970), 420-422. doi: 10.2307/2037351. Google Scholar

[32]

S. KolyadaM. Misiurewicz and L'. Snoha, Spaces of transitive interval maps, Ergodic Theory Dynam. Systems, 35 (2015), 2151-2170. doi: 10.1017/etds.2014.18. Google Scholar

[33]

S. Kolyada, M. Misiurewicz and L'. Snoha, Loops of transitive interval maps, Dynamics and numbers, Contemp. Math., Amer. Math. Soc., Providence, RI, 669 (2016), 137-154. Google Scholar

[34]

S. Kolyada and O. Rybak, On the Lyapunov numbers, Colloq. Math., 131 (2013), 209-218. doi: 10.4064/cm131-2-4. Google Scholar

[35]

S. Kolyada and J. Semikina, On topological entropy: When positivity implies +infinity, Proc. Amer. Math. Soc., 143 (2015), 1545-1558. Google Scholar

[36]

S. Kolyada and L'. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., 4 (1996), 205-233. Google Scholar

[37]

S. Kolyada and L'. Snoha, Some aspects of topological transitivity - a survey, Iteration Theory (ECIT 94) (Opava), Grazer Math. Ber., 334 (1997), 3-35. Google Scholar

[38]

S. KolyadaL'. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fund. Math., 168 (2001), 141-163. doi: 10.4064/fm168-2-5. Google Scholar

[39]

J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231. doi: 10.1016/j.topol.2011.07.013. Google Scholar

[40]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114. doi: 10.1007/s10114-015-4574-0. Google Scholar

[41]

M. Matviichuk, On the dynamics of subcontinua of a tree, J. Difference Equ. Appl., 19 (2013), 223-233. doi: 10.1080/10236198.2011.634804. Google Scholar

[42]

T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126. doi: 10.1088/0951-7715/20/9/006. Google Scholar

[43]

N. T. Nhu, The group of measure preserving transformations of the unit interval is an absolute retract, Proc. Amer. Math. Soc., 110 (1990), 515-522. doi: 10.1090/S0002-9939-1990-1009997-6. Google Scholar

[44]

K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280. doi: 10.1090/S0002-9939-1970-0250283-7. Google Scholar

[45]

P. Raith, Topological transitivity for expanding piecewise monotonic maps on the interval, Aequationes Math., 57 (1999), 303-311. doi: 10.1007/s000100050085. Google Scholar

[46]

D. Ruelle, Dynamical systems with turbulent behavior, Mathematical Problems in Theoretical Physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), Lecture Notes in Phys., vol. 80, Springer, Berlin-New York, 1978,341-360. Google Scholar

[47]

A. N. Šarkovskiĭ, On attracting and attracted sets, Soviet Math. Dokl., 6 (1965), 268-270. Google Scholar

[48]

A. N. Šarkovskiĭ, Continuous mapping on the limit points of an iteration sequence, Ukrain. Mat. Ž., 18 (1966), 127-130. Google Scholar

[49]

A. N. Šarkovskiĭ, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kiev, 1989. Google Scholar

[50]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. Google Scholar

[51]

T. Yagasaki, Weak extension theorem for measure-preserving homeomorphisms of noncompact manifolds, J. Math. Soc. Japan, 61 (2009), 687-721. doi: 10.2969/jmsj/06130687. Google Scholar

show all references

References:
[1] E. Akin, Recurrence in Topological Dynamics. Furstenberg families and Ellis actions, The University Series in Mathematics, Plenum Press, New York, 1997.
[2]

E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286. doi: 10.1007/BF02788112. Google Scholar

[3]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313. Google Scholar

[4]

E. Akin and J. Rautio, Chain transitive homeomorphisms on a space: All or none, Pacific J. Math., 291 (2017), 1-49. doi: 10.2140/pjm.2017.291.1. Google Scholar

[5]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988, Notas de Matemática [Mathematical Notes], 122. Google Scholar

[6]

J. AuslanderS. Kolyada and L'. Snoha, Functional envelope of a dynamical system, Nonlinearity, 20 (2007), 2245-2269. doi: 10.1088/0951-7715/20/9/012. Google Scholar

[7]

J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J., (2) 32 (1980), 177-188. doi: 10.2748/tmj/1178229634. Google Scholar

[8]

H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math., 60 (1967), 241-249. doi: 10.4064/fm-60-3-241-249. Google Scholar

[9]

T. DasE. Shah and L'. Snoha, (Non-)expansivity in functional envelopes, J. Math. Anal. Appl., 410 (2014), 1043-1048. doi: 10.1016/j.jmaa.2013.08.057. Google Scholar

[10]

T. Dobrowolski, Examples of topological groups homeomorphic to $l_2^f$, Proc. Amer. Math. Soc., 98 (1986), 303-311. doi: 10.2307/2045703. Google Scholar

[11]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc., (3) 4 (1954), 168-176. Google Scholar

[12]

T. Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and Topological Dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, 7-37. doi: 10.1090/conm/385/07188. Google Scholar

[13]

T. DownarowiczL'. Snoha and D. Tywoniuk, Minimal spaces with cyclic group of homeomorphisms, J. Dynam. Differential Equations, 29 (2017), 243-257. doi: 10.1007/s10884-015-9433-2. Google Scholar

[14]

F. T. Farrell and A. Gogolev, The space of Anosov diffeomorphisms, J. Lond. Math. Soc., (2) 89 (2014), 383-396. doi: 10.1112/jlms/jdt073. Google Scholar

[15]

A. Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup., 13 (1980), 45-93. doi: 10.24033/asens.1377. Google Scholar

[16]

B. R. Fayad, Topologically mixing and minimal but not ergodic, analytic transformation on ${{\rm{T}}^5}$, Bol. Soc. Brasil. Mat. (N.S.), 31 (2000), 277-285. doi: 10.1007/BF01241630. Google Scholar

[17]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49. doi: 10.1007/BF01692494. Google Scholar

[18] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.
[19]

H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math., 34 (1978), 61-85 (1979). doi: 10.1007/BF02790008. Google Scholar

[20]

P. Gartside and A. Glyn, Autohomeomorphism groups, Topology Appl., 129 (2003), 103-110. doi: 10.1016/S0166-8641(02)00140-2. Google Scholar

[21]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075. doi: 10.1088/0951-7715/6/6/014. Google Scholar

[22]

J. de Groot and R. J. Wille, Rigid continua and topological group-pictures, Arch. Math., 9 (1958), 441-446. doi: 10.1007/BF01898628. Google Scholar

[23]

J. de Groot, Groups represented by homeomorphism groups, Math. Ann., 138 (1959), 80-102. doi: 10.1007/BF01369667. Google Scholar

[24]

J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys., 70 (1979), 133-160. doi: 10.1007/BF01982351. Google Scholar

[25]

S. Harada, Remarks on the topological group of measure preserving transformations, Proc. Japan Acad., 27 (1951), 523-526. doi: 10.3792/pja/1195571228. Google Scholar

[26]

K. H. Hofmann and S. A. Morris, Compact homeomorphism groups are profinite, Topology Appl., 159 (2012), 2453-2462. doi: 10.1016/j.topol.2011.09.050. Google Scholar

[27]

K. H. Hofmann and S. A. Morris, Representing a profinite group as the homeomorphism group of a continuum, preprint, arXiv: 1108.3876.Google Scholar

[28]

W. HuangD. KhilkoS. Kolyada and G. Zhang, Dynamical compactness and sensitivity, J. Differential Equations, 260 (2016), 6800-6827. doi: 10.1016/j.jde.2016.01.011. Google Scholar

[29]

W. HuangD. KhilkoS. KolyadaA. Peris and G. Zhang, Finite intersection property and dynamical compactness, J. Dynam. Differential Equations, 30 (2018), 1221-1245. doi: 10.1007/s10884-017-9600-8. Google Scholar

[30]

W. HuangS. Kolyada and G. Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, Ergodic Theory Dynam. Systems, 38 (2018), 651-665. doi: 10.1017/etds.2016.48. Google Scholar

[31]

M. Keane, Contractibility of the automorphism group of a nonatomic measure space, Proc. Amer. Math. Soc., 26 (1970), 420-422. doi: 10.2307/2037351. Google Scholar

[32]

S. KolyadaM. Misiurewicz and L'. Snoha, Spaces of transitive interval maps, Ergodic Theory Dynam. Systems, 35 (2015), 2151-2170. doi: 10.1017/etds.2014.18. Google Scholar

[33]

S. Kolyada, M. Misiurewicz and L'. Snoha, Loops of transitive interval maps, Dynamics and numbers, Contemp. Math., Amer. Math. Soc., Providence, RI, 669 (2016), 137-154. Google Scholar

[34]

S. Kolyada and O. Rybak, On the Lyapunov numbers, Colloq. Math., 131 (2013), 209-218. doi: 10.4064/cm131-2-4. Google Scholar

[35]

S. Kolyada and J. Semikina, On topological entropy: When positivity implies +infinity, Proc. Amer. Math. Soc., 143 (2015), 1545-1558. Google Scholar

[36]

S. Kolyada and L'. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., 4 (1996), 205-233. Google Scholar

[37]

S. Kolyada and L'. Snoha, Some aspects of topological transitivity - a survey, Iteration Theory (ECIT 94) (Opava), Grazer Math. Ber., 334 (1997), 3-35. Google Scholar

[38]

S. KolyadaL'. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fund. Math., 168 (2001), 141-163. doi: 10.4064/fm168-2-5. Google Scholar

[39]

J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231. doi: 10.1016/j.topol.2011.07.013. Google Scholar

[40]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114. doi: 10.1007/s10114-015-4574-0. Google Scholar

[41]

M. Matviichuk, On the dynamics of subcontinua of a tree, J. Difference Equ. Appl., 19 (2013), 223-233. doi: 10.1080/10236198.2011.634804. Google Scholar

[42]

T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126. doi: 10.1088/0951-7715/20/9/006. Google Scholar

[43]

N. T. Nhu, The group of measure preserving transformations of the unit interval is an absolute retract, Proc. Amer. Math. Soc., 110 (1990), 515-522. doi: 10.1090/S0002-9939-1990-1009997-6. Google Scholar

[44]

K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280. doi: 10.1090/S0002-9939-1970-0250283-7. Google Scholar

[45]

P. Raith, Topological transitivity for expanding piecewise monotonic maps on the interval, Aequationes Math., 57 (1999), 303-311. doi: 10.1007/s000100050085. Google Scholar

[46]

D. Ruelle, Dynamical systems with turbulent behavior, Mathematical Problems in Theoretical Physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), Lecture Notes in Phys., vol. 80, Springer, Berlin-New York, 1978,341-360. Google Scholar

[47]

A. N. Šarkovskiĭ, On attracting and attracted sets, Soviet Math. Dokl., 6 (1965), 268-270. Google Scholar

[48]

A. N. Šarkovskiĭ, Continuous mapping on the limit points of an iteration sequence, Ukrain. Mat. Ž., 18 (1966), 127-130. Google Scholar

[49]

A. N. Šarkovskiĭ, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kiev, 1989. Google Scholar

[50]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. Google Scholar

[51]

T. Yagasaki, Weak extension theorem for measure-preserving homeomorphisms of noncompact manifolds, J. Math. Soc. Japan, 61 (2009), 687-721. doi: 10.2969/jmsj/06130687. Google Scholar

Figure 1.  The space $\mathcal{TCS}_1\cup\mathcal{TCS}_2$
Figure 2.  Box map
Figure 3.  Boxes $I_i\times I_j$
Figure 4.  Basic loop $L_2$. It consists of four arcs represented by the four rows in this picture (instead of all elements of such an arc only five of them are shown)
Figure 5.  Deformation of the 1st arc of $L_2$
Figure 6.  Deformation of the 2nd arc of $L_2$
Figure 7.  Deformation of the 3rd arc of $L_2$
Figure 8.  Deformation of the 4th arc of $L_2$
Figure 9.  From $L_2$ to auxiliary loop consisting of two arcs
Figure 10.  The 1st arc of the auxiliary loop obtained from $L_2$
Figure 11.  The 2nd arc of the auxiliary loop obtained from $L_2$
Figure 12.  Topologically transitive systems
Figure 13.  Topologically transitive, non-proximal systems
Figure 14.  The first 4 steps in the construction of the Sierpinski carpet
Figure 15.  Steps in the construction of the De Groot - Wille rigid plane continuum
Figure 16.  The Julia set for the map $z \mapsto (z^2+0.3+0.05i)/(z^2-1)$
Figure 17.  The first 5 steps in the construction of the solenoid called the Smale-Williams attractor
Figure 18.  Composant $\bar{\gamma}$ of the Slovak space
Table 1.  Transitive maps in $\mathcal{CS}_2 $
$\mathcal{CS}_2$
codepicturecondition equivalent to transitivity
$ (a, 1, 0, d) $ $ d\leq a-4 + \frac{2}{a} \quad \text{or} \quad 1-a \leq (1-d) - 4 + \frac{2}{1-d} $
$ (a, 0, 1, d) $ $ a>d $
$ (1, 0, c, d) $ $ d \leq 2 + 2c - \frac{1}{c} $
$ (a, b, 1, 0) $ $ 1-a \leq 2 + 2(1-b) - \frac{1}{1-b} $
$\mathcal{CS}_2$
codepicturecondition equivalent to transitivity
$ (a, 1, 0, d) $ $ d\leq a-4 + \frac{2}{a} \quad \text{or} \quad 1-a \leq (1-d) - 4 + \frac{2}{1-d} $
$ (a, 0, 1, d) $ $ a>d $
$ (1, 0, c, d) $ $ d \leq 2 + 2c - \frac{1}{c} $
$ (a, b, 1, 0) $ $ 1-a \leq 2 + 2(1-b) - \frac{1}{1-b} $
[1]

Alfredo Marzocchi, Sara Zandonella Necca. Attractors for dynamical systems in topological spaces. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 585-597. doi: 10.3934/dcds.2002.8.585

[2]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124

[3]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781

[4]

Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085

[5]

M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495

[6]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[7]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-19. doi: 10.3934/dcds.2019228

[8]

Mykola Matviichuk, Damoon Robatian. Chain transitive induced interval maps on continua. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 741-755. doi: 10.3934/dcds.2015.35.741

[9]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086

[10]

Liangming Chen, Ming Cao, Chuanjiang Li. Bearing rigidity and formation stabilization for multiple rigid bodies in $ SE(3) $. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 257-267. doi: 10.3934/naco.2019017

[11]

Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451

[12]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[13]

Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715

[14]

Abdelwahab Bensouilah, Sahbi Keraani. Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2961-2976. doi: 10.3934/dcds.2019123

[15]

Alain Bensoussan, Miroslav Bulíček, Jens Frehse. Existence and compactness for weak solutions to Bellman systems with critical growth. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1729-1750. doi: 10.3934/dcdsb.2012.17.1729

[16]

H.T. Banks, S. Dediu, H.K. Nguyen. Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space. Mathematical Biosciences & Engineering, 2007, 4 (3) : 403-430. doi: 10.3934/mbe.2007.4.403

[17]

Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467

[18]

Thomas French. Follower, predecessor, and extender set sequences of $ \beta $-shifts. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4331-4344. doi: 10.3934/dcds.2019175

[19]

María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations & Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018

[20]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (13)
  • HTML views (220)
  • Cited by (0)

Other articles
by authors

[Back to Top]