Sergii Kolyada (∗December 7th, 1957, † May 16th, 2018) was a distinguished Ukrainian mathematician. He worked in low-dimensional dynamical systems and in topological dynamics. He studied dynamical systems (X, f ) given by a (usually compact metric) space X and a continuous map f : X→X.
If we want to describe his work in keywords, then the following ones are probably the most appropriate: zero Schwarzian, triangular map, ω-limit set, topological entropy, minimal map, minimal set, chaos, sensitivity, functional envelope of a dynamical system, dynamical compactness.
He introduced, together with his coauthors, several new notions: functional envelope of a dynamical system, Li-Yorke sensitivity, dynamical topology, dynamical compactness.
SergiI Kolyada started with interval dynamics. He studied dynamics of one-parameter families of continuousmaps (among others, the problemofmonotonicity of topological entropy) and dynamics of continuous piecewise zero-Schwarzian maps. For instance he proved that unimodal maps of this class have, under some assumptions, at most one attracting periodic orbit (due to Singer, such a property had been known for three times differentiable unimodal maps with negative Schwarzian).
After this interval period, he concentrated mostly on topological dynamics and low dimensional dynamics. We will not speak on all his results. However, we are going to describe those of them which he liked most or which are, in our opinion, most important.
1. Low dimensional dynamical systems
1.1. Triangular maps. A triangular map is a continuous map F : I²→I², where I = [0;1], of the form
F(x;y) = (f (x);g(x;y)).
Such a map preserves ‘vertical’ fibres, the fibre over x is mapped to a fibre over f (x). So, it is a special skew product. Triangular maps became popular in 1979, when Kloeden [3] proved that Sharkovsky theorem worked for them. Then it was natural to believe that in fact most of the results from interval dynamics can be carried over to them (to check this, is sometimes called Sharkovsky’s program). Kolyada was the first who showed that this belief was false. In [4] he developed the basics of the theory of triangular maps. In particular, he constructed a triangular map of type 2∞ with positive topological entropy (recall that interval maps of type 2∞ have zero entropy; a map is of type 2∞ if it has periodic orbits of periods all powers of two, and of no other periods).
1.2. ω-limit sets. Given a point x in a dynamical system (X, f), its ω-limit set ωf(x) is the set of all limit points of the trajectory x, f (x), f²(x),… . Topological structure of ω-limit sets depends on the phase space X. As established in [1], a nonempty closed subset M of I = [0;1] is an ω-limit set for some continuous map f : I→I if and only if M is nowhere dense or a union of finitely many nondegenerate closed intervals. If M is a subset of a vertical fibre in the square I², it can be an ω-limit set of a triangular map F of the square even if the topological structure of M is more complicated. The reason is that now the point x with ωF(x) = M can be chosen outside that fibre. A full topological characterization of ω-limit sets of triangular maps which lie in just one vertical fibre, was found, in co-authorship,
by Kolyada in [5]. The result has an interesting consequence we are going to describe.
Due to Dowker and Friedlander [2] for homeomorphisms and Sharkovsky [10] for continuous maps, it is known that for dynamical systems on compact metric spaces the following are equivalent:
(1) (X; f ) can be embedded as an ω-limit set in some larger system (Y,g),
(2) (X; f ) is f – connected (meaning that there is no nonempty, proper, closed set
A⊆X such that f (A)⊆IntA).
Now, the above mentioned result from [5] implies the following: A nonempty closed subset X of the unit interval is an ω-limit set (i.e., there is a dynamical system (Y,g) containing X as an ω-limit set, or equivalently, X admits a continuous selfmap f such that (X, f) is f -connected) if and only if X is not a disjoint union of a finite number of nondegenerate closed intervals and a nonempty countable set whose distance from at least one of those intervals is
positive.
2. Minimality
2.1. Minimal dynamical systems. In topological dynamics, the most fundamental dynamical systems are the minimal ones. These are systems which have no nontrivial subsystems. More precisely, a dynamical system (X,f) is called minimal if X does not contain any non-empty, proper, closed f-invariant set (a set M⊆X is f-invariant if f(M)⊆M). In such a case we also say that the map f itself is minimal. An equivalent definition is: (X, f) is minimal if for every x∈X, the orbit {x, f(x), f²(x),…} is dense in X.
Minimality was one of favorite topics of Sergi˘ ı and he contributed to this area significantly. With coauthors he showed that if (X, f) is minimal with X a compact metric space, then in many aspects the continuous map f behaves like a homeomorhism [8]:
- there is no nonempty redundant open set for f (G⊆X is said to be redundant for f if f (X\G) = f (X));
- f is feebly open (i.e. it sends nonempty open sets to sets with non-empty interior);
- f preserves the topological size of a set in both directions. More precisely,
the implication; - f is almost 1-to-1.
Further, in [8] also the existence of minimal noninvertible maps on the torus has been established. This was the first example of a minimal noninvertible map on a manifold.
2.2. Minimal sets. Given a dynamical system (X, f), a set M⊆X is called a minimal set if it is non-empty, closed and f-invariant and no proper subset of M has these three properties. So, a nonempty closed set M⊆X is a minimal set if and only if (M, f|Μ) is a minimal system. A system (X, f) is minimal if and only if X is the (unique) minimal set in (X, f). The basic fact discovered by G. D. Birkhoff is that in any compact system (X, f) there are minimal sets.
Minimal sets are fundamental objects of study in topological dynamics. A big open problem is: How do minimal sets look like? Two major contributions of Kolyada and his coauthors to this problem are the following results.
The first of themis that forminimal sets on 2-manifolds the dichotomy “nowhere dense or everything” holds. More precisely, in [9] it is proved that on compact connected 2-manifolds with or without boundary, aminimal set either is the whole manifold or is nowhere dense.
The second contribution is a very detailed description of minimal sets of continuous fibre-preserving maps in graph bundles. The complete results are too long to describe here, but one particular corollary says that the fibre-preserving maps in tree bundles have only nowhere dense minimal sets. In particular, if F is a continuous triangular map in the square I² and M is a minimal set of F, then M is nowhere dense in the space pr1(M)×I (pr1 denotes the projection onto the first factor; the nowhere density in the square I² is trivial, but here pr1(M)×I is a very
small subspace of the square, since pr1(M) is a Cantor set or a finite set). Moreover, either a typical fibre of M is a Cantor set, or there is a positive integer N such that a typical fibre of M has cardinality N.
3. Dynamical topology
Let us recall that in Topological Dynamics one investigates dynamical properties that can be described in topological terms. Say, topological transitivity is a dynamical property of a map and is defined in terms of behaviour of open sets under the iterates of the map.
In contrast with this, in [6] and [7] Kolyada and his coauthors introduced the notion of Dynamical Topology. It is the area where one investigates topological properties of spaces of maps that can be described in dynamical terms. For instance, transitivity of maps is a dynamical property. So, one can ask what are the topological properties of the space of all transitive maps on a given space. One of the results from [6] says that the space of all transitive maps on the interval [0;1] is contractible.
References
[1] S.J. Agronsky, A.M. Bruckner, J.G. Ceder and T.L. Pearson, The structure of ω-limit sets for continuous functions, Real Analysis Exchange 15 (1989-90), 483–510.
[2] Y. N. Dowker, F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc. (3) 4 (1954), 168–176.
[3] P.E. Kloeden, On Sharkovsky’s cycle coexistence ordering, Bull. Austral. Math. Soc. 20 (1979), no. 2, 171–177.
[4] S. F. Kolyada, On dynamics of triangular maps of the square, Ergodic Theory Dynam. Systems 12 (1992), no. 4, 749–768.
[5] S. F. Kolyada, L’. Snoha, On ω-limit sets of triangular maps, Real Anal. Exchange 18 (1992/93), no. 1, 115–130.
[6] S. Kolyada, M. Misiurewicz, L’. Snoha, Spaces of transitive interval maps, Ergodic Theory Dynam. Systems 35 (2015), no. 7, 2151–2170.
[7] S. Kolyada, M. Misiurewicz, L’. Snoha, Loops of transitive interval maps. Dynamics and numbers, 137–154, Contemp. Math. 669, Amer. Math. Soc., Providence, RI, 2016.
[8] S. Kolyada, L’. Snoha, S. Trofimchuk, Noninvertible minimal maps, Fund. Math. 168 (2001),
no. 2, 141–163.
[9] S. Kolyada, L’. Snoha, S. Trofimchuk, Proper minimal sets on compact connected 2-manifolds are nowhere dense, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 863–876.
[10] O. M. Sharkovski, On attracting and attracted sets (Russian), Dokl. Akad. Nauk SSSR
160 (1965), 1036–1038. Translated in Soviet Math. Dokl. 6 (1965), 268–270.
This is a part of a longer text written by prof. LUBOMIR SNOHA
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Slovakia.
The list of other important Kolyada’s works.
- S. Kolyada, L’. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), no. 2-3, 205–233.
- AN Sharkovskiı, SF Kolyada, AG Sivak, VV Fedorenko, Dynamics of one-dimensional maps, 1997, Kluwer Academic Publishing.
- S. Kolyada, M. Misiurewicz, L’. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval, Fund. Math. 160 (1999), no. 2, 161–181.
- F. Blanchard, E. Glasner, S. Kolyada, A. Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547 (2002), 51–68.
- E. Akin, S. Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003), no. 4, 1421–1433.
- W. Huang, D. Khilko, S. Kolyada, Guohua Zhang, Dynamical compactness and sensitivity, J. Differential Equations 260 (2016), no. 9, 6800–6827.
- W. Huang, D. Khilko, S. Kolyada, A. Peris, Guohua Zhang, Finite intersection property and dynamical compactness, J. Dynam. Diff. Equations, published online on 27 June 2017, doi:10.1007/s10884-017-9600-8, pp.1–25.