Grzegorz Swiatek
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Grzegorz Swiatek has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Genadi Levin
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
12 shared publications Jacek Graczyk
University of Paris
11 shared publications- Janina Kotus
Warsaw University of Technology
4 shared publications Duncan Sands
University of Paris
3 shared publications- Jacek Graczyk
University of Paris XI
2 shared publications Dmitry Khavinson
1 shared publication
Latest Publications
- 28 September 2020
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 41 (8) , 1-52
Limit drift for complex Feigenbaum mappings
We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to under the dynamics of the tower for corresponding . That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when tends to . We also prove the convergence of the drifts to a finite limit, which can be expressed purely in terms of the limiting tower, which corresponds to a Feigenbaum map with a flat critical point.
- 28 September 2020
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 41 (8) , 1-52
Limit drift for complex Feigenbaum mappings
We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to under the dynamics of the tower for corresponding . That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when tends to . We also prove the convergence of the drifts to a finite limit, which can be expressed purely in terms of the limiting tower, which corresponds to a Feigenbaum map with a flat critical point.
- Open Access
Limit Drift for Complex Feigenbaum Mappings
We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order \ell of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to 0 under the dynamics of the tower for corresponding \ell. That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when \ell tends to infinity. We also prove the convergence of the drifts to a finite limit which can be expressed purely in terms of the limiting tower which corresponds to a Feigenbaum map with a flat critical point
Open AccessLimit Drift for Complex Feigenbaum Mappings
We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order \ell of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to 0 under the dynamics of the tower for corresponding \ell. That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when \ell tends to infinity. We also prove the convergence of the drifts to a finite limit which can be expressed purely in terms of the limiting tower which corresponds to a Feigenbaum map with a flat critical point
- Open Access
- 1 January 2017
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 37 (5) , 2717-2743
Diagonal stationary points of the bethe functional
We investigate stationary points of the Bethe functional for the Ising model on a $2$-dimensional lattice. Such stationary points are also fixed points of message passing algorithms. In the absence of an external field, by symmetry reasons one expects the fixed points to have constant means at all sites. This is shown not to be the case. There is a critical value of the coupling parameter which is equal to the phase transition parameter on the computation tree, see [13], above which fixed points appear with means that are variable though constant on diagonals of the lattice and hence the term “diagonal stationary points”. A rigorous analytic proof of their existence is presented. Furthermore, computer-obtained examples of diagonal stationary points which are local maxima of the Bethe functional and hence stable equilibria for message passing are shown. The smallest such example was found on the$100× 100$ lattice.- 1 January 2017
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 37 (5) , 2717-2743
Open AccessDiagonal stationary points of the bethe functional
We investigate stationary points of the Bethe functional for the Ising model on a $2$-dimensional lattice. Such stationary points are also fixed points of message passing algorithms. In the absence of an external field, by symmetry reasons one expects the fixed points to have constant means at all sites. This is shown not to be the case. There is a critical value of the coupling parameter which is equal to the phase transition parameter on the computation tree, see [13], above which fixed points appear with means that are variable though constant on diagonals of the lattice and hence the term “diagonal stationary points”. A rigorous analytic proof of their existence is presented. Furthermore, computer-obtained examples of diagonal stationary points which are local maxima of the Bethe functional and hence stable equilibria for message passing are shown. The smallest such example was found on the$100× 100$ lattice. - 13 August 2016
- Published by Springer Nature in Mathematische Annalen
- Vol. 369 (1-2) , 49-108
Fine structure of connectedness loci
- 13 August 2016
- Published by Springer Nature in Mathematische Annalen
- Vol. 369 (1-2) , 49-108
Fine structure of connectedness loci
- 4 July 2016
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 37 (8) , 2643-2670
Limit drift
We study the problem of the existence of wild attractors for critical circle coverings with Fibonacci dynamics. This is known to be related to the drift for the corresponding fixed points of renormalization. The fixed point depends only on the order of the critical point $\ell$ and its drift is a number $\unicode[STIX]{x1D717}(\ell )$ which is finite for each finite $\ell$ . We show that the limit $\unicode[STIX]{x1D717}(\infty ):=\lim _{\ell \rightarrow \infty }\unicode[STIX]{x1D717}(\ell )$ exists and is finite. The finiteness of the limit is in a sharp contrast with the case of Fibonacci unimodal maps. Furthermore, $\unicode[STIX]{x1D717}(\infty )$ is expressed as a contour integral in terms of the limit of the fixed points of renormalization when $\ell \rightarrow \infty$ . There is a certain paradox here, since this dynamical limit is a circle homeomorphism with the golden mean rotation number whose own drift is $\infty$ for topological reasons.- 4 July 2016
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 37 (8) , 2643-2670
Limit drift
We study the problem of the existence of wild attractors for critical circle coverings with Fibonacci dynamics. This is known to be related to the drift for the corresponding fixed points of renormalization. The fixed point depends only on the order of the critical point $\ell$ and its drift is a number $\unicode[STIX]{x1D717}(\ell )$ which is finite for each finite $\ell$ . We show that the limit $\unicode[STIX]{x1D717}(\infty ):=\lim _{\ell \rightarrow \infty }\unicode[STIX]{x1D717}(\ell )$ exists and is finite. The finiteness of the limit is in a sharp contrast with the case of Fibonacci unimodal maps. Furthermore, $\unicode[STIX]{x1D717}(\infty )$ is expressed as a contour integral in terms of the limit of the fixed points of renormalization when $\ell \rightarrow \infty$ . There is a certain paradox here, since this dynamical limit is a circle homeomorphism with the golden mean rotation number whose own drift is $\infty$ for topological reasons.- Open Access
Limit drift
We study the problem of the existence of wild attractors for critical circle covering maps with Fibonacci dynamics.
Open AccessLimit drift
We study the problem of the existence of wild attractors for critical circle covering maps with Fibonacci dynamics.
- Open Access
- 1 April 2013
- Published by International Press of Boston in Arkiv för Matematik
- Vol. 51 (1) , 53-69
Asymptotic porosity of planar harmonic measure
Project Euclid - mathematics and statistics online
- 1 April 2013
- Published by International Press of Boston in Arkiv för Matematik
- Vol. 51 (1) , 53-69
Open AccessAsymptotic porosity of planar harmonic measure
Project Euclid - mathematics and statistics online
- 5 May 2012
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 312 (3) , 695-734
Common Limits of Fibonacci Circle Maps
- 5 May 2012
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 312 (3) , 695-734
Common Limits of Fibonacci Circle Maps
- Open Access
Common Limits of Fibonacci Circle Maps
We show that limits for the critical exponent tending to \infty exist in both critical circle homeomorphisms of golden mean rotation number and Fibonacci circle coverings. Moreover, they are the same. The limit map is not analytic at the critical point, which is flat, but has non-trivial complex dynamics.
Open AccessCommon Limits of Fibonacci Circle Maps
We show that limits for the critical exponent tending to \infty exist in both critical circle homeomorphisms of golden mean rotation number and Fibonacci circle coverings. Moreover, they are the same. The limit map is not analytic at the critical point, which is flat, but has non-trivial complex dynamics.
- 29 June 2009
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 30 (3) , 855-875
Measure of the Julia set of the Feigenbaum map with infinite criticality
We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.
- 29 June 2009
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 30 (3) , 855-875
Measure of the Julia set of the Feigenbaum map with infinite criticality
We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.