## Periodic Reporting for period 1 - ShadOmIC (Shadowing, Omega Limit Sets and Internal Chain Transitivity in Dynamical Systems)

Reporting period: 2015-08-30 to 2017-08-29

Imagine a system that evolves over time, for example the economy, the weather or the positions of the planets in the solar system. Some systems of this kind, for example the solar system, are deterministic in the sense that the state of the system now completely determines the state of the system in, say a year's time. One way of modelling such systems is to consider a function f mapping the space X of all possible states to itself.

In this approach, if the system is in the state x at time 0, then at time 1 it is in the state f(x), at time 2 it is in the state f(f(x))=f^2(x) and so on. Systems of this kind are called discrete dynamical systems. Typically, we assume that f is

a continuous function from X to itself and that X is a compact metric space, we refer to the states in X as points. In studying such systems one is often interested in how a particular point (or state) develops over time, so that one is interested in the orbit of the point, that is the sequence of points x, f(x), f^2(x), f^3(x), and so on. One might also be interested in the the limiting behaviour of a point x, i.e. the set of points to which the orbit converges, which is a measure of how the system behaves in the long term. In our example of the solar system, for example, given the current configuration of the solar system, we might be interested in whether in the future it gradually flies apart or collapse in to the sun or remains in some steady state.

Now suppose that we are trying to calculate the orbit of a point, for example using a computer. The computer will introduce rounding errors, so that each time we try to calculate f(x) we obtain a value that is close to, but not exactly equal to f(x). This leads us to the concept of a 'd-psuedo-orbits', that is a sequence of points x_1, x_2, x_3, and so on, with the property that the distance between f(x_n) (the actual value of x_n after one time step) and x_{n+1} (the computer calculated value) is less than d. It turns out that in some dynamical systems one can 'shadow' pseudo-orbits by real orbits, so that we know that the computed pseudo-orbit is not too far away from an actual orbit. Technically we say that f has the shadowing property provided that for any (potentially very small) e>0, there is a d>0 such that for every d-pseudo-orbit x_1, x_2, x_3, ... , there is a point z such that the distance from the nth point in the pseudo-orbit and the nt point in the orbit of z is no greater than e. This notion of shadowing has played an important role in the theory of dynamical systems. Furthermore, in an omega-limit set, for any d, there is a d-pseudo-orbit between any two points, which is to say that omega limit sets are 'internally chain transitive'.

The notion of shadowing is closely related to characterizations of omega-limit sets in terms of internal chain transitivity and the aim of the ShadOmIC project is to develop the general theory behind these relations. The project contributes to the state of the art in an important area of mathematics, the theory of dynamical systems. It brings a leading young mathematician from the US to Europe, establishing new international collaborations and enhancing the reputation and research capacity of the European Union.

In this approach, if the system is in the state x at time 0, then at time 1 it is in the state f(x), at time 2 it is in the state f(f(x))=f^2(x) and so on. Systems of this kind are called discrete dynamical systems. Typically, we assume that f is

a continuous function from X to itself and that X is a compact metric space, we refer to the states in X as points. In studying such systems one is often interested in how a particular point (or state) develops over time, so that one is interested in the orbit of the point, that is the sequence of points x, f(x), f^2(x), f^3(x), and so on. One might also be interested in the the limiting behaviour of a point x, i.e. the set of points to which the orbit converges, which is a measure of how the system behaves in the long term. In our example of the solar system, for example, given the current configuration of the solar system, we might be interested in whether in the future it gradually flies apart or collapse in to the sun or remains in some steady state.

Now suppose that we are trying to calculate the orbit of a point, for example using a computer. The computer will introduce rounding errors, so that each time we try to calculate f(x) we obtain a value that is close to, but not exactly equal to f(x). This leads us to the concept of a 'd-psuedo-orbits', that is a sequence of points x_1, x_2, x_3, and so on, with the property that the distance between f(x_n) (the actual value of x_n after one time step) and x_{n+1} (the computer calculated value) is less than d. It turns out that in some dynamical systems one can 'shadow' pseudo-orbits by real orbits, so that we know that the computed pseudo-orbit is not too far away from an actual orbit. Technically we say that f has the shadowing property provided that for any (potentially very small) e>0, there is a d>0 such that for every d-pseudo-orbit x_1, x_2, x_3, ... , there is a point z such that the distance from the nth point in the pseudo-orbit and the nt point in the orbit of z is no greater than e. This notion of shadowing has played an important role in the theory of dynamical systems. Furthermore, in an omega-limit set, for any d, there is a d-pseudo-orbit between any two points, which is to say that omega limit sets are 'internally chain transitive'.

The notion of shadowing is closely related to characterizations of omega-limit sets in terms of internal chain transitivity and the aim of the ShadOmIC project is to develop the general theory behind these relations. The project contributes to the state of the art in an important area of mathematics, the theory of dynamical systems. It brings a leading young mathematician from the US to Europe, establishing new international collaborations and enhancing the reputation and research capacity of the European Union.

Good, the Principal Investigator, and Meddaugh, the Fellow, worked closely on the detailed research questions outlined in the project proposal.

The first phase of the project was to study the relationship between omega-limit sets and internally chain transitive sets in the presence of shadowing. Surprisingly, we were able to give a complete picture in terms of variants of the notion of shadowing, which in large part also answered the general question behind the proposed third phase of the project. This work has been accepted for publication

The second phase of the project aimed to characterize shadowing or demonstrate its presence in particular families of dynamical systems. Meddaugh, with collaborators from the US, demonstrated that shadowing is generic in one particular family of systems, the dendrites. This result has been accepted for publication. However, during the course of work on this second phase of the project, we began to suspect that a totally general characterization of shadowing might be possible in terms of classical systems called shifts of finite type. This is indeed turned out to be the case, resulting in a significant new contribution to the theory of dynamical systems. This result has been submitted for publication. Some of the theory used in this general characterization of shadowing was developed in part by Good and a collaborator form Mexico. Again these results have been accepted for publication. Other questions relating to shadowing in specific families originally associated with the second phase of the project remain unanswered.

During the last few months of the project, Good and Meddaugh returned to outstanding questions from the first and third phases related to alpha-limit sets, rather than omega-limit sets. We were expecting this to be a short study with some relatively easily obtained results, closely mirroring the results obtained for omega-limit sets. The relationship between shadowing alpha-limit sets and internal transitivity turns out to be more complex and interesting than we first thought and this work is ongoing.

The project ended before any significant progress was made on the final highly speculative phase of the proposed research.

In addition to the three published papers and one preprint so far arising out of the project, Good and Meddaugh have given a number of seminars and conference talks on the results obtained during the project, both in the UK and internationally.

The first phase of the project was to study the relationship between omega-limit sets and internally chain transitive sets in the presence of shadowing. Surprisingly, we were able to give a complete picture in terms of variants of the notion of shadowing, which in large part also answered the general question behind the proposed third phase of the project. This work has been accepted for publication

The second phase of the project aimed to characterize shadowing or demonstrate its presence in particular families of dynamical systems. Meddaugh, with collaborators from the US, demonstrated that shadowing is generic in one particular family of systems, the dendrites. This result has been accepted for publication. However, during the course of work on this second phase of the project, we began to suspect that a totally general characterization of shadowing might be possible in terms of classical systems called shifts of finite type. This is indeed turned out to be the case, resulting in a significant new contribution to the theory of dynamical systems. This result has been submitted for publication. Some of the theory used in this general characterization of shadowing was developed in part by Good and a collaborator form Mexico. Again these results have been accepted for publication. Other questions relating to shadowing in specific families originally associated with the second phase of the project remain unanswered.

During the last few months of the project, Good and Meddaugh returned to outstanding questions from the first and third phases related to alpha-limit sets, rather than omega-limit sets. We were expecting this to be a short study with some relatively easily obtained results, closely mirroring the results obtained for omega-limit sets. The relationship between shadowing alpha-limit sets and internal transitivity turns out to be more complex and interesting than we first thought and this work is ongoing.

The project ended before any significant progress was made on the final highly speculative phase of the proposed research.

In addition to the three published papers and one preprint so far arising out of the project, Good and Meddaugh have given a number of seminars and conference talks on the results obtained during the project, both in the UK and internationally.

"The project has resulted in three papers accepted for publication in leading journals (Ergodic Theory and Dynamical Systems, Discrete and Continuous Dynamical Systems) and one preprint. Two of these represent significant advances in the state of the art of our understanding of dynamical systems. The paper ""Orbital Shadowing, Internal Chain Transitivity and ω-limit sets"" characterizes when internally chain transitive sets are omega-limit sets in terms of a variant of the notion of shadowing that is closely related to notions used in stability theory, raising the possibility of a more topological approach to stability. In the preprint ""Shifts of finite type as fundamental objects in the theory of shadowing"", systems with shadowing are shown to be built up in a very natural way from shifts of finite type, classical objects that, for example, play a key role in work of Bowen and Smale."