arXiv subjects

Jeremias Epperlein

Publications and source records attributed to Jeremias Epperlein.

14 recordsLinked to original sources

arXiv2025

Lower Bounds for the Shadiness Constant of Finite-Dimensional Normed Spaces

By the Hahn-Banach theorem, every normed space admits rank-one projections with operator norm one. However, this is not true for higher rank projections. Bosznay and Garay showed that for every $d \geq 3$ there exist $d$-dimensional normed spaces $X$ for which all projections of rank $k$, with $2 \leq k \leq d-1$, have norm larger than or equal to some constant $c>1$. We call the maximal such constant the shadiness constant of $X$. Although constructing such spaces is not difficult, few explicit estimates of their shadiness constants exist. We show how optimization techniques can provide provable lower bounds for these shadiness constants. As an application, we construct a $3$-dimensional normed space whose unit ball is a polytope with $12$ vertices, with shadiness constant at least $1.01$. Furthermore we show that there is no shady norm on $\mathbb{R}^3$ whose unit ball is a polytope with $10$ or fewer vertices, thereby confirming a conjecture by Bosznay and Garay.

math.FA↗

arXiv2025

Auerbach bases, projection constants, and the joint spectral radius of principal submatrices

It is shown that compact sets of complex matrices can always be brought, via similarity transformation, into a form where all matrix entries are bounded in absolute value by the joint spectral radius (JSR). The key tool for this is that every extremal norm of a matrix set admits an Auerbach basis; any such basis gives rise to a desired coordinate system. An immediate implication is that all diagonal entries - equivalently, all one-dimensional principal submatrices - are uniformly bounded above by the JSR. It is shown that the corresponding bounding property does not hold for higher dimensional principal submatrices. More precisely, we construct finite matrix sets for which, across the entire similarity orbit, the JSRs of all higher-dimensional principal submatrices exceed that of the original set. This shows that the bounding result does not extend to submatrices of dimension greater than one. The constructions rely on tools from the geometry of finite-dimensional Banach spaces, with projection constants of norms playing a key role. Additional bounds of the JSR of principal submatrices are obtained using John's ellipsoidal approximation and known estimates for projection constants.

math.DS↗

arXiv2024

Hyperspatiality for isomorphisms of stabilized automorphism groups of shifts of finite type

Given a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the stabilized automorphism group $\textrm{Aut}^{\infty}(T)$ of the system $(X,T)$ is the group of self-homeomorphisms of $X$ which commute with some power of $T$. We study the question of spatiality for stabilized automorphism groups of shifts of finite type. We prove that any isomorphism $Ψ\colon \textrm{Aut}^{\infty}(σ_{m}) \to \textrm{Aut}^{\infty}(σ_{n})$ between stabilized automorphism groups of full shifts is spatially induced by a homeomorphism $\hatΨ$ between respective stabilized spaces of chain recurrent subshifts. This spatialization in particular gives a bijection between the sets of periodic points which intertwines some powers of the shifts, and this bijection recovers the isomorphism at the level of the faithful actions on the sets of periodic points. We also prove that the outer automorphism group of $\textrm{Aut}^{\infty}(σ_{n})$ is uncountable, and deduce several other properties of $\textrm{Aut}^{\infty}(σ_{n})$ using the spatiality results.

math.DS↗

arXiv2023

The joint spectral radius is pointwise H\"older continuous

We show that the joint spectral radius is pointwise H\"older continuous. In addition, the joint spectral radius is locally H\"older continuous for $\varepsilon$-inflations. In the two-dimensional case, local H\"older continuity holds on the matrix sets with positive joint spectral radius.

math.DS↗

arXiv2023

Eventual Conjugacy of Free Inert $G$-SFTs

The action of a finite group $G$ on a subshift of finite type $X$ is called free, if every point has trivial stabilizer, and it is called inert, if the induced action on the dimension group of $X$ is trivial. We show that any two free inert actions of a finite group $G$ on an SFT are conjugate by an automorphism of any sufficiently high power of the shift space. This partially answers a question posed by Fiebig. As a consequence we obtain that every two free elements of the stabilized automorphism group of a full shift are conjugate in this group. In addition, we generalize a result of Boyle, Carlsen and Eilers concerning the flow equivalence of $G$-SFTs.

math.DS↗

arXiv2022

On Game Theory Using Stochastic Tail Orders

We consider a family of distributions on which natural tail orders can be constructed upon a representation of a distribution by a (single) hyper-real number. Past research revealed that the ordering can herein strongly depend on the particular model of the hyperreals, specifically the underlying ultrafilter. Hence, our distribution family is constructed to order invariantly of an ultrafilter. Moreover, we prove that it lies dense in the set of all distributions with the (same) compact support, w.r.t. the supremum norm. Overall, this work resents a correction to [10, 12], in response to recent findings of [2].

math.PR↗

arXiv2022

Remarks on the tail order on moment sequences

We consider positively supported Borel measures for which all moments exist. On the set of compactly supported measures in this class a partial order is defined via eventual dominance of the moment sequences. Special classes are identified on which the order is total, but it is shown that already for the set of distributions with compactly supported smooth densities the order is not total. In particular we construct a pair of measures with smooth density for which infinitely many moments agree and another one for which the moments alternate infinitely often. This disproves some recently published claims to the contrary. Some consequences for games with distributional payoffs are discussed.

math.CA↗

arXiv2020

Iterated Minkowski sums, horoballs and north-south dynamics

Given a finite generating set $A$ for a group $Γ$, we study the map $W \mapsto WA$ as a topological dynamical system -- a continuous self-map of the compact metrizable space of subsets of $Γ$. If the set $A$ generates $Γ$ as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when $Γ= \mathbb{Z}^d$ and $A \subseteq \mathbb{Z}^d$ a finite positively generating set containing the natural invertible extension of the map $W \mapsto W+A$ is always topologically conjugate to the unique "north-south" dynamics on the Cantor set. In contrast to this, we show that various natural "geometric" properties of the finitely generated group $(Γ,A)$ can be recovered from the dynamics of this map, in particular, the growth type and amenability of $Γ$. When $Γ= \mathbb{Z}^d$, we show that the volume of the convex hull of the generating set $A$ is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group $Γ$, related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.

math.DS↗

arXiv2019

Wagoner's Complexes Revisited

We generalize Wagoner's representation of the automorphism group of a two-sided subshifts of finite type as the fundamental group of a certain CW-complex to groupoids having a certain refinement structure. This significantly streamlines the original proof and allows us to extend this construction to, e.g., the automorphism group of subshifts of finite type over arbitrary finitely generated groups and the automorphism group of G-SFTs.

math.DS↗

arXiv2018

On Arbitrarily Long Periodic Orbits of Evolutionary Games on Graphs

A periodic behavior is a well observed phenomena in biological and economical systems. We show that evolutionary games on graphs with imitation dynamics can display periodic behavior for an arbitrary choice of game theoretical parameters describing social-dilemma games. We construct graphs and corresponding initial conditions whose trajectories are periodic with an arbitrary minimal period length. We also examine a periodic behavior of evolutionary games on graphs with the underlying graph being an acyclic (tree) graph. Astonishingly, even this acyclic structure allows for arbitrary long periodic behavior.

math.DS↗

arXiv2015

Mixing properties in coded systems

We show that topological mixing, weak mixing and total transitivity are equivalent for coded systems. We provide an example of a mixing coded system which cannot be approximated by any increasing sequence of mixing shifts of finite type, has only periodic points of even period and each set of its generators consists of blocks of even length. We prove that such an example cannot be a synchronized system. We also show that a mixing coded systems has the strong property $P$.

math.DS↗

arXiv2014

Evolutionary Games on Graphs and Discrete Dynamical Systems

Evolutionary games on graphs play an important role in the study of evolution of cooperation in applied biology. Using rigorous mathematical concepts from a dynamical systems and graph theoretical point of view, we formalize the notions of attractor, update rules and update orders. We prove results on attractors for different utility functions and update orders. For complete graphs we characterize attractors for synchronous and sequential update rules. In other cases (for $k$-regular graphs or for different update orders) we provide sufficient conditions for attractivity of full cooperation and full defection. We construct examples to show that these conditions are not necessary. Finally, by formulating a list of open questions we emphasize the advantages of our rigorous approach.

math.DS↗

arXiv2012

Signature stability analysis for networks of coupled dynamical systems with Hermitian Jacobian

The central theme of complex systems research is understanding the emergent macroscopic properties of a system from the interplay of its microscopic constituents. Here, we ask what conditions a complex network of microscopic dynamical units has to meet to permit stationary macroscopic dynamics, such as stable equilibria or phase-locked states. We present an analytical approach which is based on a graphical notation that allows rewriting Jacobi's signature criterion in an interpretable form. The derived conditions pertain to topological structures on all scales, ranging from individual nodes to the interaction network as a whole. Our approach can be applied to many systems of symmetrically coupled units. For the purpose of illustration, we consider the example of synchronization, specifically the (heterogeneous) Kuramoto model and an adaptive variant. Moreover, we discuss how the graphical notation can be employed to study isospectrality in Hermitian matrices. The results complete and extend the previous analysis of Do et al. [Phys. Rev. Lett. 108, 194102 (2012)].

math.DS↗

arXiv2012

Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian

A linear system $\dot x = Ax$, $A \in \mathbb{R}^{n \times n}$, $x \in \mathbb{R}^n$, with $\mathrm{rk} A = n-1$, has a one-dimensional center manifold $E^c = \{v \in \mathbb{R}^n : Av=0\}$. If a differential equation $\dot x = f(x)$ has a one-dimensional center manifold $W^c$ at an equilibrium $x^*$ then $E^c$ is tangential to $W^c$ with $A = Df(x^*)$ and for stability of $W^c$ it is necessary that $A$ has no spectrum in $\mathbb{C}^+$, i.e.\ if $A$ is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to $A$, we formulate meso-scale conditions with certain principal minors of $A$ which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.

math.DS↗