## Final Report Summary - UNIVERSALITY (Universality in TopologicalDynamics)

Universality is a classical and fundamental subject in dimension theory. The Marie Curie CIG project "Universality" aims to investigate the phenomenon of universality in topological dynamics through the invariant of mean dimension. Topological dynamics is a subfield of the theory of dynamical systems. Examples of dynamical systems abound in nature (the solar system, movement of gas molecules, the flow of water in rivers etc.). In topological dynamics certain aspects of such systems are investigated. An effective way of gaining understanding of the phenomena possible is through embedding into universal spaces. A system is called universal for a family of systems if all the members of the family may be embedded into this particular system. Finding meaningful universal spaces may be achieved with the help of mean dimension, an invariant which heuristically measures the growth rate of dimension as times evolves. Mean dimension was introduced by Gromov in 1999 and has already found applications in diverse fields such as symbolic dynamics, cellular automata, holomorphic functions and mathematical physics as well as topological dynamics, notably in the Boyle-Downarowicz symbolic extension entropy theorem and its generalization by the Researcher to the context of several commuting transformations. The far reaching Lindenstrauss-Tsukamoto conjecture is essentially the claim that mean dimension and periodic dimension are the only obstructions for embedding an arbitrary dynamical system into well understood models - shifts over Euclidean cubes.

One of the achievements of the Project has been establishing together with Lei Jin and Masaki Tsukamoto of refinements of the classical Bebutov-Kakutani theorem from 1968 by finding compact universal spaces for families of real flows. In addition, a "bottom-up" approach towards characterization of dynamical systems was developed with Eli Glasner and XiangDong Ye by exhibiting concrete non-trivial new factors for all minimal dynamical systems, leading to an answer to a question posed by Fields medalist Terence Tao.

The project also tackled conjectures involving the small boundary property, a dynamical analogue of vanishing covering dimension, and the topological Rokhlin property, a topological analogue of the Rokhlin Lemma from measured dynamics. A surprising connection to mathematical physics was discovered by showing that a certain model of the two-dimensional Navier-Stokes equations from fluid mechanics can be embedded into a cubical shift. It was also shown that the celebrated Takens theorem may be quantitatively improved if one allows continuous observables.

A highlight of the Project has been the solution together with Masaki Tsukamoto of a problem posed by Fields medalist Elon Lindenstrauss in 1999 about optimal embedding of minimal systems into shifts over Euclidean cubes. The solution has a fascinating feature. The nature of the statement itself is purely in topological dynamics. But crucial ingredients in the proof are Fourier analysis and complex function theory. Therefore a new unexpected link has been forged between topological dynamics and classical analysis.

Another apex of the Project has been the solution together with Elon Lindenstrauss and Masaki Tsukamoto of a problem open for about 15 years regarding embedding into cubical shifts in the context of several commuting transformations. Assuming a strengthening of the topological Rokhlin property, namely the so-called marker property, it was shown for an action by several commuting transformations, that vanishing mean dimension is equivalent to representation by an inverse limit of finite entropy systems as well as to the small boundary property. In a subsequent work with Yixiao Qiao and Masaki Tsukamoto the embedding result was improved and an optimal result was achieved for minimal systems for several commuting transformations into shifts over Euclidean cubes. The proof uses mutli-dimensional signal analysis and differs fundamentally from the proof for actions by a a single transformation.

Research was performed in line with the research objectives detailed in Part B of Annex I of the grant agreement. The project has resulted in five articles already published or accepted for publication and five articles submitted for publication.

At the professional level the project aimed in integrating the Researcher into the Host Institute. This goal has been achieved as the Researcher has been awarded a long-term position and has obtained the habilitaion degree. Presently, the Researcher is conducting independent research and collaborating internationally. He has co-advised one M.Sc. student, co-advised two PhD students who have recently submitted their theses and currently is mentoring a first year PhD student. The Researcher has engaged in dissemination activities with respect to the project to graduate students, fellow mathematicians and the public at large.

By its quality, originality and novelty the project contributed to EU excellence, as well as to the Host Institute, the mathematical community and the public, through transfer of knowledge, cooperation and dissemination. Moreover, some of the results of the project were used by mathematicians working in the field of C*-algerbas of noncommutative geometry, helping to create a bridge between two distinct disciplines of mathematics. List of publications of the Project:

1. Mean dimension & Jaworski-type theorems. Proceedings of the London Mathematical Society, 111(4):831-850, 2015.

2. Takens embedding theorem with a continuous observable. In Ergodic Theory: Advances in Dynamical Systems, pp. 134-142. Walter de Gruyter GmbH & Co KG, 2016.

3. Mean dimension of Z^k-actions. Joint with Elon Lindenstrauss and Masaki Tsukamoto. Geom. Funct. Anal. 26(3):778–817, 2016.

4. An explicit compact universal space for real flows. Joint with Lei Jin. Submitted 2016.

5. Embedding topological dynamical systems with periodic points in cubical shifts. Ergodic Theory Dynam. Systems 37 (2017), 512–538.

6. Embedding minimal dynamical systems into Hilbert cubes. Joint with Masaki Tsukamoto. Submitted 2017.

7. Application of signal analysis to the embedding problem of Zk-actions. Joint with Yixiao Qiao and Masaki Tsukamoto. Submitted 2017.

8. Higher order regionally proximal equivalence relations for general minimal group actions. Joint with Eli Glasner and XiangDong Ye. Submitted 2017.

9. A Lipschitz refinement of the Bebutov-Kakutani dynamical embedding theorem. Joint with Lei Jin and Masaki Tsukamoto. Submitted 2017.

10. The embedding problem in topological dynamics and Takens' theorem. Joint with Yixiao Qiao and Gábor Szabó. To appear in Nonlinearity, 2017.

One of the achievements of the Project has been establishing together with Lei Jin and Masaki Tsukamoto of refinements of the classical Bebutov-Kakutani theorem from 1968 by finding compact universal spaces for families of real flows. In addition, a "bottom-up" approach towards characterization of dynamical systems was developed with Eli Glasner and XiangDong Ye by exhibiting concrete non-trivial new factors for all minimal dynamical systems, leading to an answer to a question posed by Fields medalist Terence Tao.

The project also tackled conjectures involving the small boundary property, a dynamical analogue of vanishing covering dimension, and the topological Rokhlin property, a topological analogue of the Rokhlin Lemma from measured dynamics. A surprising connection to mathematical physics was discovered by showing that a certain model of the two-dimensional Navier-Stokes equations from fluid mechanics can be embedded into a cubical shift. It was also shown that the celebrated Takens theorem may be quantitatively improved if one allows continuous observables.

A highlight of the Project has been the solution together with Masaki Tsukamoto of a problem posed by Fields medalist Elon Lindenstrauss in 1999 about optimal embedding of minimal systems into shifts over Euclidean cubes. The solution has a fascinating feature. The nature of the statement itself is purely in topological dynamics. But crucial ingredients in the proof are Fourier analysis and complex function theory. Therefore a new unexpected link has been forged between topological dynamics and classical analysis.

Another apex of the Project has been the solution together with Elon Lindenstrauss and Masaki Tsukamoto of a problem open for about 15 years regarding embedding into cubical shifts in the context of several commuting transformations. Assuming a strengthening of the topological Rokhlin property, namely the so-called marker property, it was shown for an action by several commuting transformations, that vanishing mean dimension is equivalent to representation by an inverse limit of finite entropy systems as well as to the small boundary property. In a subsequent work with Yixiao Qiao and Masaki Tsukamoto the embedding result was improved and an optimal result was achieved for minimal systems for several commuting transformations into shifts over Euclidean cubes. The proof uses mutli-dimensional signal analysis and differs fundamentally from the proof for actions by a a single transformation.

Research was performed in line with the research objectives detailed in Part B of Annex I of the grant agreement. The project has resulted in five articles already published or accepted for publication and five articles submitted for publication.

At the professional level the project aimed in integrating the Researcher into the Host Institute. This goal has been achieved as the Researcher has been awarded a long-term position and has obtained the habilitaion degree. Presently, the Researcher is conducting independent research and collaborating internationally. He has co-advised one M.Sc. student, co-advised two PhD students who have recently submitted their theses and currently is mentoring a first year PhD student. The Researcher has engaged in dissemination activities with respect to the project to graduate students, fellow mathematicians and the public at large.

By its quality, originality and novelty the project contributed to EU excellence, as well as to the Host Institute, the mathematical community and the public, through transfer of knowledge, cooperation and dissemination. Moreover, some of the results of the project were used by mathematicians working in the field of C*-algerbas of noncommutative geometry, helping to create a bridge between two distinct disciplines of mathematics. List of publications of the Project:

1. Mean dimension & Jaworski-type theorems. Proceedings of the London Mathematical Society, 111(4):831-850, 2015.

2. Takens embedding theorem with a continuous observable. In Ergodic Theory: Advances in Dynamical Systems, pp. 134-142. Walter de Gruyter GmbH & Co KG, 2016.

3. Mean dimension of Z^k-actions. Joint with Elon Lindenstrauss and Masaki Tsukamoto. Geom. Funct. Anal. 26(3):778–817, 2016.

4. An explicit compact universal space for real flows. Joint with Lei Jin. Submitted 2016.

5. Embedding topological dynamical systems with periodic points in cubical shifts. Ergodic Theory Dynam. Systems 37 (2017), 512–538.

6. Embedding minimal dynamical systems into Hilbert cubes. Joint with Masaki Tsukamoto. Submitted 2017.

7. Application of signal analysis to the embedding problem of Zk-actions. Joint with Yixiao Qiao and Masaki Tsukamoto. Submitted 2017.

8. Higher order regionally proximal equivalence relations for general minimal group actions. Joint with Eli Glasner and XiangDong Ye. Submitted 2017.

9. A Lipschitz refinement of the Bebutov-Kakutani dynamical embedding theorem. Joint with Lei Jin and Masaki Tsukamoto. Submitted 2017.

10. The embedding problem in topological dynamics and Takens' theorem. Joint with Yixiao Qiao and Gábor Szabó. To appear in Nonlinearity, 2017.