arXiv subjects

M. Stadlbauer

Publications and source records attributed to M. Stadlbauer.

6 recordsLinked to original sources

arXiv2021

Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

We show the existence of invariant ergodic $σ$-additive probability measures with full support on $X$ for a class of linear operators $L: X \to X$, where $L$ is a weighted shift operator and $X$ either is the Banach space $c_0(\mathbb{R})$ or $l^p(\mathbb{R})$ for $1\leq p<\infty$. In order to do so, we adapt ideas from Thermodynamic Formalism as follows. For a given bounded Hölder continuous potential $A:X \to \mathbb{R}$, we define a transfer operator $\mathcal{L}_A$ which acts on continuous functions on $X$ and prove that this operator satisfies a Ruelle-Perron-Frobenius theorem. That is, we show the existence of an eigenfunction for $\mathcal{L}_A$ which provides us with a normalized potential $\overline{A}$ and an action of the dual operator $\mathcal{L}_{\overline{A}}^*$ on the $1$-Wasserstein space of probabilities on $X$ with a unique fixed point, to which we refer to as Gibbs probability. It is worth noting that the definition of $\mathcal{L}_A$ requires an {\it a priori} probability on the kernel of $L$. These results are extended to a wide class of operators with a non-trivial kernel defined on separable Banach spaces.

math.DS↗

arXiv2019

Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures. One of the main results of the article is to show that a probability is DLR-Gibbs (associated to a continuous translation invariant specification), if and only if, is an eigenprobability for the transpose of the Ruelle operator. Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present

math.DS↗

arXiv2019

Thermodynamic Formalism for Topological Markov Chains on Borel Standard Spaces

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space $X\equiv E^{\mathbb{N}}$, where $E$ is a general Borel standard space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on $X$ and obtain the existence of equilibrium states as additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states do not have a purely additive part. We then apply our results to the construction of invariant measures of time-homogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite dimensional diffusion.

math.DS↗

arXiv2018

Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples

In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalize the Perron-Frobenius-Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined $\zeta$-function at its critical parameter. We then generalize Sharp's construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the $\zeta$-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.

math.DS↗

arXiv2016

Spectral Properties of the Ruelle Operator for Product Type Potentials on Shift Spaces

We study a class of potentials $f$ on one sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness property of these quantities is also discussed and it is shown that there always exists a Bernoulli equilibrium state even if $f$ does not satisfy Bowen's condition. We apply these results to potentials $f:\{-1,1\}^\mathbb{N} \to \mathbb{R}$ of the form $$ f(x_1,x_2,\ldots) = x_1 + 2^{-\gamma} \, x_2 + 3^{-\gamma} \, x_3 + ...+n^{-\gamma} \, x_n + \ldots $$ with $\gamma >1$. For $3/2 < \gamma \leq 2$, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist a.s. (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.

math.DS↗

arXiv2007

Investigation of the chemical vicinity of crystal defects in ion-irradiated Mg and AZ31 with coincident Doppler broadening spectroscopy

Crystal defects in magnesium and magnesium based alloys like AZ31 are of major importance for the understanding of their macroscopic properties. We have investigated defects and their chemical surrounding in Mg and AZ31 on an atomic scale with Doppler broadening spectroscopy of the positron annihilation radiation. In these Doppler spectra the chemical information and the defect contribution have to be thoroughly separated. For this reason samples of annealed Mg were irradiated with Mg-ions in order to create exclusively defects. In addition Al- and Zn-ion irradiation on Mg-samples was performed in order to create samples with defects and impurity atoms. The ion irradiated area on the samples was investigated with laterally and depth resolved positron Doppler broadening spectroscopy (DBS) and compared with preceding SRIM-simulations of the vacancy distribution, which are in excellent agreement. The investigation of the chemical vicinity of crystal defects in AZ31 was performed with coincident Doppler broadening spectroscopy (CDBS) by comparing Mg-ion irradiated AZ31 with Mg-ion irradiated Mg. No formation of solute-vacancy complexes was found due to the ion irradiation, despite the high defect mobility.

cond-mat.mtrl-sci↗