Final Report Summary - DS-LOWDIM (Dynamical Systems in Low Dimensions)
The completed project was in the area of the theory of dynamical systems which belongs to most multi-disciplined areas of present-day mathematics, being also among most active ones. By a dynamical system, we understand a topological space endowed with a continuous action of a (semi-)group of continuous transformations on the space.
The (semi)-group determines whether we work with discrete or continuous time. In the discrete-time case, we distinguish two cases:
- the semigroup is generated by all the forward iterates of a continuous, in general noninvertible map of the space, or
- we have a group generated by all the forward and backward iterates of a homeomorphism of the space.
In the continuous-time, case we consider a (semi-)flow, i.e. an action of the real (half-)line on the space.
The project focused on several topics in the dynamical systems theory which can be roughly sorted into two groups according to their nature:
- minimality, transitivity, quasi-periodically forced systems,
- dynamical zeta functions, topological entropy, topological pressure.
Minimality, transitivity, quasi-periodically forced systems
A system is (topologically) minimal if all orbits are dense. Minimality is a central topic in topological dynamics - minimal systems are the most fundamental ones being topologically irreducible. In the discrete-time case, we distinguish two notions of minimality: a map is minimal if all forward orbits are dense, and a homeomorphism is weakly minimal if all full orbits are dense. These two notions coincide for homeomorphisms on compact spaces (Gottschalk, 1944). We have completely analogous situation in the case of flows (see [2]). A weaker property is (topological) transitivity which means that there is at least one dense orbit in the system.
The most basic question is about the existence of (weakly) minimal systems - what spaces admit weakly minimal homeomorphisms and/or minimal maps? In the case of one-manifolds (and more generally topological graphs), it is known that only a disjoint union of circles admits a minimal map which, in turn, must be a homeomorphism (Balibrea, Hric and Snoha, 2003). The answer in the case of two-manifolds was given by Blokh, Oversteegen and Tymchatyn (2005): only a disjoint union of tori and a disjoint union of Klein bottles admit minimal maps. The situation in higher dimensions has not yet been explored. In [1], we constructed a new rich class of minimal systems with very complicated behaviour - almost totally disconnected minimal systems. We also fully classified minimal sets on these spaces and proved several other general results; as a consequence, we got e.g. a full topological classification of minimal sets on dendrites.
In [2], we look at problems related to dense orbits in a very general topological setting. We prove several general results from which we point out a quite useful one relating minimality and transitivity of continuous-time and discrete-time systems - from a continuous-time systems, we easily construct a discrete-time one by looking at a t-map, the map determined by where the system gets by the flow in time t. Even if the flow is minimal, this does not automatically hold for the t-map - it depends on the choice of the time t. It was known that, for compact spaces, there are residually many times which preserve minimality (Fayad, 2000). We prove this for general not necessarily compact systems and, moreover, we prove that such times can be chosen such that the orbit of a point is dense under flow if and only if it is also under the t-map. One of the motivations to work on this topic was also to fix a few common misunderstandings concerning minimality and transitivity.
Closely related to minimality is the topic of quasi-periodically forced systems. In [3], we study quasi-periodically forced systems on the annulus. We show how to construct almost automorphic minimal sets in a simple way contrary to the known complicated approach. We construct an explicit example of embedded Denjoy dynamics in a system with no invariant curves. We prove other related resutls in this setting.
Dynamical zeta functions, topological entropy, topological pressure
Study of periodic orbits has always been one of central topics in dynamical systems theory. Counting periodic orbits is also a natural task from the point of view of ergodic theory since the simplest invariant measures for a dynamical system are those carried by periodic orbits. Dynamical zeta functions are an effective tool for this purpose. Moreover, they comprehend a lot of complex information about the underlying systems in a manner of a simple complex function. In several classes of systems, zeta functions determine also directly the value of topological entropy. The first one was introduced by Artin and Mazur in 1965 which was later modified in 1988 by Milnor and Thurston to better suit the setting of piecewise monotone maps. We continue in this direction using the completely new approach (Alvesand Ramos, 1999). Using this, in [5], we study the local growth of periodic orbits and show some interesting properties. In [6], we transfer the techniques originally developed for piecewise monotone interval maps to subshifts of infinite type. While there is a more-less direct way how to do this for subshifts of finite type, the infinite type is much more delicate and it is not possible to apply the known techniques directly.
Sometimes it is necessary to use sequential tools to detect more subtle features of a system. One of examples is detecting chaos by topological sequence entropy. Analogous but more general tool is topological sequence pressure, which we developed in [4]. We treat basic properties of this notion - we prove the ones which hold and provide counterexamples for the others which do not carry from the nonsequential case. Connection from topological pressure to zeta functions is provided by the weighted case.
[1] F. Balibrea, T. Downarowicz, R. Hric, L. Snoha and V. pitalský: Almost totally disconnected minimal systems. Ergodic Theory and Dynamical Systems 29 (2009), 737 - 766
[2] R. Hric and L. Snoha: Dense orbits and misunderstandings around them. Submitted 2012
[3] R. Hric and T. Jäger: A simple construction of almost automorphic minimal sets. Preprint 2012
[4] R. Hric and G. Iommi: Topological sequence pressure. Preprint 2012
[5] J. Alves and R. Hric: Local growth of periodic orbits. Preprint 2012
[6] J. Alves, R. Hric and M. Málek: Zeta functions: from piecewise monotone maps to subshifts of infinite type. Preprint 2012.
The (semi)-group determines whether we work with discrete or continuous time. In the discrete-time case, we distinguish two cases:
- the semigroup is generated by all the forward iterates of a continuous, in general noninvertible map of the space, or
- we have a group generated by all the forward and backward iterates of a homeomorphism of the space.
In the continuous-time, case we consider a (semi-)flow, i.e. an action of the real (half-)line on the space.
The project focused on several topics in the dynamical systems theory which can be roughly sorted into two groups according to their nature:
- minimality, transitivity, quasi-periodically forced systems,
- dynamical zeta functions, topological entropy, topological pressure.
Minimality, transitivity, quasi-periodically forced systems
A system is (topologically) minimal if all orbits are dense. Minimality is a central topic in topological dynamics - minimal systems are the most fundamental ones being topologically irreducible. In the discrete-time case, we distinguish two notions of minimality: a map is minimal if all forward orbits are dense, and a homeomorphism is weakly minimal if all full orbits are dense. These two notions coincide for homeomorphisms on compact spaces (Gottschalk, 1944). We have completely analogous situation in the case of flows (see [2]). A weaker property is (topological) transitivity which means that there is at least one dense orbit in the system.
The most basic question is about the existence of (weakly) minimal systems - what spaces admit weakly minimal homeomorphisms and/or minimal maps? In the case of one-manifolds (and more generally topological graphs), it is known that only a disjoint union of circles admits a minimal map which, in turn, must be a homeomorphism (Balibrea, Hric and Snoha, 2003). The answer in the case of two-manifolds was given by Blokh, Oversteegen and Tymchatyn (2005): only a disjoint union of tori and a disjoint union of Klein bottles admit minimal maps. The situation in higher dimensions has not yet been explored. In [1], we constructed a new rich class of minimal systems with very complicated behaviour - almost totally disconnected minimal systems. We also fully classified minimal sets on these spaces and proved several other general results; as a consequence, we got e.g. a full topological classification of minimal sets on dendrites.
In [2], we look at problems related to dense orbits in a very general topological setting. We prove several general results from which we point out a quite useful one relating minimality and transitivity of continuous-time and discrete-time systems - from a continuous-time systems, we easily construct a discrete-time one by looking at a t-map, the map determined by where the system gets by the flow in time t. Even if the flow is minimal, this does not automatically hold for the t-map - it depends on the choice of the time t. It was known that, for compact spaces, there are residually many times which preserve minimality (Fayad, 2000). We prove this for general not necessarily compact systems and, moreover, we prove that such times can be chosen such that the orbit of a point is dense under flow if and only if it is also under the t-map. One of the motivations to work on this topic was also to fix a few common misunderstandings concerning minimality and transitivity.
Closely related to minimality is the topic of quasi-periodically forced systems. In [3], we study quasi-periodically forced systems on the annulus. We show how to construct almost automorphic minimal sets in a simple way contrary to the known complicated approach. We construct an explicit example of embedded Denjoy dynamics in a system with no invariant curves. We prove other related resutls in this setting.
Dynamical zeta functions, topological entropy, topological pressure
Study of periodic orbits has always been one of central topics in dynamical systems theory. Counting periodic orbits is also a natural task from the point of view of ergodic theory since the simplest invariant measures for a dynamical system are those carried by periodic orbits. Dynamical zeta functions are an effective tool for this purpose. Moreover, they comprehend a lot of complex information about the underlying systems in a manner of a simple complex function. In several classes of systems, zeta functions determine also directly the value of topological entropy. The first one was introduced by Artin and Mazur in 1965 which was later modified in 1988 by Milnor and Thurston to better suit the setting of piecewise monotone maps. We continue in this direction using the completely new approach (Alvesand Ramos, 1999). Using this, in [5], we study the local growth of periodic orbits and show some interesting properties. In [6], we transfer the techniques originally developed for piecewise monotone interval maps to subshifts of infinite type. While there is a more-less direct way how to do this for subshifts of finite type, the infinite type is much more delicate and it is not possible to apply the known techniques directly.
Sometimes it is necessary to use sequential tools to detect more subtle features of a system. One of examples is detecting chaos by topological sequence entropy. Analogous but more general tool is topological sequence pressure, which we developed in [4]. We treat basic properties of this notion - we prove the ones which hold and provide counterexamples for the others which do not carry from the nonsequential case. Connection from topological pressure to zeta functions is provided by the weighted case.
[1] F. Balibrea, T. Downarowicz, R. Hric, L. Snoha and V. pitalský: Almost totally disconnected minimal systems. Ergodic Theory and Dynamical Systems 29 (2009), 737 - 766
[2] R. Hric and L. Snoha: Dense orbits and misunderstandings around them. Submitted 2012
[3] R. Hric and T. Jäger: A simple construction of almost automorphic minimal sets. Preprint 2012
[4] R. Hric and G. Iommi: Topological sequence pressure. Preprint 2012
[5] J. Alves and R. Hric: Local growth of periodic orbits. Preprint 2012
[6] J. Alves, R. Hric and M. Málek: Zeta functions: from piecewise monotone maps to subshifts of infinite type. Preprint 2012.