Gabriel Fuhrmann
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View article: On the probability of positive finite-time Lyapunov exponents on strange nonchaotic attractors
On the probability of positive finite-time Lyapunov exponents on strange nonchaotic attractors Open
We study strange non-chaotic attractors in a class of quasiperiodically forced monotone interval maps known as pinched skew products. We prove that the probability of positive time-$ N $ Lyapunov exponents—with respect to the unique physic…
View article: Tame or wild Toeplitz shifts
Tame or wild Toeplitz shifts Open
We investigate tameness of Toeplitz shifts. By introducing the notion of extended Bratteli–Vershik diagrams, we show that such shifts with finite Toeplitz rank are tame if and only if there are at most countably many orbits of singular fib…
View article: On the continuity of Følner averages
On the continuity of Følner averages Open
It is known that if each point $x$ of a dynamical system is generic for some invariant measure $μ_x$, then there is a strong connection between certain ergodic and topological properties of that system. In particular, if the acting group i…
View article: Amorphic complexity of group actions with applications to quasicrystals
Amorphic complexity of group actions with applications to quasicrystals Open
In this article, we define amorphic complexity for actions of locally compact -compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for -actions, is a topological invariant which measures the complex…
View article: On the effect of forcing on fold bifurcations and early-warning signals in population dynamics
On the effect of forcing on fold bifurcations and early-warning signals in population dynamics Open
The classical fold bifurcation is a paradigmatic example of a critical transition. It has been used in a variety of contexts, including in particular ecology and climate science, to motivate the role of slow recovery rates and increased au…
View article: On the probability of positive finite-time Lyapunov exponents on strange non-chaotic attractors
On the probability of positive finite-time Lyapunov exponents on strange non-chaotic attractors Open
We study strange non-chaotic attractors in a class of quasiperiodically forced monotone interval maps known as pinched skew products. We prove that the probability of positive time-N Lyapunov exponents, with respect to the unique physical …
View article: The structure of mean equicontinuous group actions
The structure of mean equicontinuous group actions Open
We study mean equicontinuous actions of locally compact σ-compact amenable groups on compact metric spaces. In this setting, we establish the equivalence of mean equicontinuity and topo-isomorphy to the maximal equicontinuous factor and pr…
View article: The bifurcation set as a topological invariant for one-dimensional dynamics
The bifurcation set as a topological invariant for one-dimensional dynamics Open
For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of (some of) their endpoints. By assuming a globa…
View article: Irregular model sets and tame dynamics
Irregular model sets and tame dynamics Open
We study the dynamical properties of irregular model sets and show that the translation action on their hull always admits an infinite independence set. The dynamics can therefore not be tame and the topological sequence entropy is strictl…
View article: Amorphic complexity of group actions with applications to quasicrystals
Amorphic complexity of group actions with applications to quasicrystals Open
In this article, we define amorphic complexity for actions of locally compact $σ$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb Z$-actions, is a topological invariant which measur…
View article: Tame or wild Toeplitz shifts
Tame or wild Toeplitz shifts Open
We investigate tameness of Toeplitz shifts. By introducing the notion of extended Bratteli-Vershik diagrams, we show that such shifts with finite Toeplitz rank are tame if and only if there are at most countably many orbits of singular fib…
View article: Constant length substitutions, iterated function systems and amorphic complexity
Constant length substitutions, iterated function systems and amorphic complexity Open
We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of…
View article: On the effect of forcing of fold bifurcations and early-warning signals in population dynamics
On the effect of forcing of fold bifurcations and early-warning signals in population dynamics Open
The classical fold bifurcation is a paradigmatic example of a critical transition. It has been used in a variety of contexts, including in particular ecology and climate science, to motivate the role of slow recovery rates and increased au…
View article: The bifurcation set as a topological invariant for one-dimensional\n dynamics
The bifurcation set as a topological invariant for one-dimensional\n dynamics Open
For a continuous map on the unit interval or circle, we define the\nbifurcation set to be the collection of those interval holes whose surviving\nset is sensitive to arbitrarily small changes of their position. By assuming a\nglobal perspe…
View article: The structure of mean equicontinuous group actions
The structure of mean equicontinuous group actions Open
We study mean equicontinuous actions of locally compact $σ$-compact amenable groups on compact metric spaces. In this setting, we establish the equivalence of mean equicontinuity and topo-isomorphy to the maximal equicontinuous factor and …
View article: Non-smooth saddle-node bifurcations II: Dimensions of strange attractors
Non-smooth saddle-node bifurcations II: Dimensions of strange attractors Open
We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates o…
View article: Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent
Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent Open
We study order-preserving $\mathcal{C}^1$-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergo…
View article: Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent
Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent Open
We study order-preserving C^1-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures…
View article: Amorphic complexity
Amorphic complexity Open
We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For inst…