# American Institute of Mathematical Sciences

## 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. Auslander, S. 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. Das, E. 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. Downarowicz, L'. 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. Huang, D. Khilko, S. 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. Huang, D. Khilko, S. Kolyada, A. 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. Huang, S. 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. Kolyada, M. 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. Kolyada, L'. 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. Auslander, S. 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. Das, E. 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. Downarowicz, L'. 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. Huang, D. Khilko, S. 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. Huang, D. Khilko, S. Kolyada, A. 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. Huang, S. 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. Kolyada, M. 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. Kolyada, L'. 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
The space $\mathcal{TCS}_1\cup\mathcal{TCS}_2$
Box map
Boxes $I_i\times I_j$
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)
Deformation of the 1st arc of $L_2$
Deformation of the 2nd arc of $L_2$
Deformation of the 3rd arc of $L_2$
Deformation of the 4th arc of $L_2$
From $L_2$ to auxiliary loop consisting of two arcs
The 1st arc of the auxiliary loop obtained from $L_2$
The 2nd arc of the auxiliary loop obtained from $L_2$
Topologically transitive systems
Topologically transitive, non-proximal systems
The first 4 steps in the construction of the Sierpinski carpet
Steps in the construction of the De Groot - Wille rigid plane continuum
The Julia set for the map $z \mapsto (z^2+0.3+0.05i)/(z^2-1)$
The first 5 steps in the construction of the solenoid called the Smale-Williams attractor
Composant $\bar{\gamma}$ of the Slovak space
Transitive maps in $\mathcal{CS}_2$
 $\mathcal{CS}_2$ code picture condition equivalent to transitivity $(a, 1, 0, d)$ $d\leq a-4 + \frac{2}{a} \quad \text{or} \quad 1-a \leq (1-d) - 4 + \frac{2}{1-d}$ $(a, 0, 1, d)$ $a>d$ $(1, 0, c, d)$ $d \leq 2 + 2c - \frac{1}{c}$ $(a, b, 1, 0)$ $1-a \leq 2 + 2(1-b) - \frac{1}{1-b}$
 $\mathcal{CS}_2$ code picture condition equivalent to transitivity $(a, 1, 0, d)$ $d\leq a-4 + \frac{2}{a} \quad \text{or} \quad 1-a \leq (1-d) - 4 + \frac{2}{1-d}$ $(a, 0, 1, d)$ $a>d$ $(1, 0, c, d)$ $d \leq 2 + 2c - \frac{1}{c}$ $(a, b, 1, 0)$ $1-a \leq 2 + 2(1-b) - \frac{1}{1-b}$
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