# Karl-Olof Lindahl

Associate ProfessorHello! I am Associate Professor in Mathematics at LNU. Before my present position at LNU I also worked as postdoc at Brown Univeristy, Pontífica Universidad Católica de Chile and Universidad de Santiago de Chile.

At the moment I teach

- Linear Algebra
- Applied probability and statistics
- Numerical methods
- Research methodology for mathematical sciences
- P-adic analysis

I am also main supervisor for a PhD student, Jonas Nordqvist.

Research projects: Dynamical Systems, Number Theory, Algebra, Discrete Mathematics and Statistics.

International collaborators are Juan Rivera-Letelier (University of Rochester), Mike Zieve (Univ. of Michigan), Charles Favre (École Polytechnique, Paris). Youssef Fares (Universite de Picardie).

Before my present position at LNU I did postdocs at Brown Univeristy, Pontífica Universidad Católica de Chile and Universidad de Santiago de Chile.

I have also worked as guest reseracher at Université de Picardie, Amiens,

**Examples of potential topics for PhD and degree projects**

1. Periodic points and nonlinear phenomena in arithmetical dynamical systems (ADS).

Periodic phenomena or cycles arise in many systems that evolve with time; physical, biological, financial and technical. The theory of dynamical systems concerns mathematical models of such systems. Of particular interest is characterization of cycles and the border between periodic, quasi-periodic, and non-periodic behaviour.

In the simplest mathematical models the dynamics is modeled by function iteration; a given state of the system is transformed into a new state by a function. From the work of Henri Poincaré in the late nineteenth century it is known that such models can mimic the behaviour of very complex systems.

As a model, we focus on ADS where the evolution in time is modeled by iterations of functions defined over fields like the rational numbers, p-adic fields, fields of positive characteristic, or complex numbers. Proposed research questions concern linearizability near periodic points, normal forms, and geometric distribution of periodic points.

Outcomes of expected results of the project could serve as the foundation of a geometric approach for solving Poonen's conjecture, stating

that quadratic polynomials with rational coefficients cannot have rational periodic points of exact period greater than three. The conjecture is a fundamental case of the Morton-Silverman conjecture that is driving the field of ADS.

2. Computational algebra and dynamical systems over finite rings.

Since the introduction of the BBS pseudorandom number generator in the 1980's there has been an increasing interest in dynamical systems defined by function iteration over finite rings. Other motivations is security in ICT systems and Pollard's rho-algorithm for prime factorization.

Given a map f over a finite ring we can associate with f a directed graph over the elements in the ring. We propose to study periodic points, the number of invariant components of the corresponding graph, and statistical aspects of such systems. This will involve tools from computational algebra, (analytic) number theory, graph theory, statistics and probability theory.

Even if patterns in the dynamics of simple maps are often irregular, averages over different primes can have more regular behaviour. The project could also include statistical aspects including comparison between function iteration over finite fields and the statistics of random maps.

- Member of the board of the faculty of technology (FTK)
- Coordinator for degree projects in mathematics

### Publications

#### Article in journal (Refereed)

- Lindahl, K., Rivera-Letelier, J. (2016). Generic parabolic points are isolated in positive characteristic.
*Nonlinearity*. 29. 1596-1621. - Lindahl, K., Rivera-Letelier, J. (2016). Optimal cycles in ultrametric dynamics and minimally ramified power series.
*Compositio Mathematica*. 152. 187-222. - Lindahl, K. (2013). The size of quadratic
*p*-adic linearization disks.*Advances in Mathematics*. 248. 872-894. - Lindahl, K., Zieve, M. (2010). On Hyperbolic Fixed Points in Ultrametric Dynamics.
*P-Adic Numbers, Ultrametric Analysis, and Applications*. 2. 232-240. - Lindahl, K. (2010). Divergence and convergence of conjugacies in non-Archimedean dynamics.
*Contemporary Mathematics*. 508. 89-109. - Lindahl, K. (2010). Applied Algebraic Dynamics.
*P-Adic Numbers, Ultrametric Analysis, and Applications*. 2. 360-362. - Lindahl, K. (2009). Linearization in Ultrametric Dynamics in Fields of Characteristic Zero — Equal Characteristic Case.
*P-Adic Numbers, Ultrametric Analysis, and Applications*. 1. 307-316. - Lindahl, K. (2004). On Siegel's linearization theorem for fields of prime characteristic.
*Nonlinearity*. 17. 745-763. - Lindahl, K., Gundlach, M., Khrennikov, A. (2001). On Ergodic Behavior of p-adic dynamical systems.
*Infin. Dimens. Anal. Quantum Probab. Relat. Top.*. 4. 569-577. - Albeverio, S., Gundlach, M., Khrennikov, A., Lindahl, K. (2001). On the Markovian Behavior of p-Adic Random Dynamical Systems.
*Russ. J. Math. Phys.*. 8. 135-152.

#### Conference paper (Refereed)

- Lindahl, K. (2002). Some results on convergence of conjugating functions over non-Archimedean fields.
*Dynamical systems from number theory to physics - 2*. 67-77.

#### Doctoral thesis, monograph (Other academic)

- Lindahl, K. (2007).
*On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point*. Doctoral Thesis. Växjö, Växjö University Press. 160.

#### Manuscript (preprint) (Other academic)

#### Article, book review (Other (popular science, discussion, etc.))

- Lindahl, K. (2006).
*Uniqueness for p-adic meromorphic products*.*Mathematical Reviews*.