Spectral triples and Dixmier trace representations of Gibbs measures: theory and examples L Cioletti, L Y Hataishi, A O Lopes, M Stadlbauer Nonlinearity, 2024 In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalise the Perron–Frobenius–Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined ζ-function at its critical parameter. We then generalise 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 ζ-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.
Quenched and annealed equilibrium states for random Ruelle expanding maps and applications MANUEL STADLBAUER, PAULO VARANDAS, XUAN ZHANG Ergodic Theory and Dynamical Systems, 2023 We find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by Baladi [Correlation spectrum of quenched and annealed equilibrium states for random expanding maps. Comm. Math. Phys.186 (1997), 671–700] and Carvalho et al [Semigroup actions of expanding maps. J. Stat. Phys.116(1) (2017), 114–136], where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact. We give applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and introduce a boundary of equilibria for not necessarily free semigroup actions.
Invariant probabilities for discrete time linear dynamics via thermodynamic formalism Artur O Lopes, Ali Messaoudi, Manuel Stadlbauer, Victor Vargas Nonlinearity, 2021 We show the existence of invariant ergodic σ-additive probability measures with full support on X for a class of linear operators L : X → X, where L is a weighted shift operator and X either is the Banach space c 0 ( R ) or l p ( R ) for 1 ⩽ p < ∞. In order to do so, we adapt ideas from thermodynamic formalism as follows. For a given bounded Hölder continuous potential A : X → R , we define a transfer operator 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 L A which provides us with a normalised potential A ¯ and an action of the dual operator L A ¯ * on the one-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 L A requires an 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.
On Spaceability within Linear Dynamics M Saavedra, M Stadlbauer arXiv preprint arXiv:2509.06156 , 2025 2025
Spectral triples and Dixmier trace representations of Gibbs measures: theory and examples L Cioletti, LY Hataishi, AO Lopes, M Stadlbauer Nonlinearity 37 , 2024 2024 Citations: 8
Minimal distance between random orbits S Gouëzel, J Rousseau, M Stadlbauer Probability Theory and Related Fields 189 (3), 811-847 , 2024 2024 Citations: 9
Amenable graphs and the spectral radius of extensions of Markov maps J Jaerisch, E Rocha, M Stadlbauer arXiv preprint arXiv:2404.08270 , 2024 2024
Quasi-rigid operators and hyper-recurrence M Saavedra, M Stadlbauer arXiv preprint arXiv:2403.17904 , 2024 2024 Citations: 1
Quenched and annealed equilibrium states for random Ruelle expanding maps and applications M Stadlbauer, P Varandas, X Zhang Ergodic Theory and Dynamical Systems 43 (9), 3150-3192 , 2023 2023 Citations: 11
The Martin boundary of an extension by a hyperbolic group SRP Bispo, M Stadlbauer Israel Journal of Mathematics 255 (1), 1-62 , 2023 2023 Citations: 1
Invariant probabilities for discrete time linear dynamics via thermodynamic formalism AO Lopes, A Messaoudi, M Stadlbauer, V Vargas Nonlinearity 34 (12), 8359-8391 , 2021 2021 Citations: 7
Thermodynamic formalism for random non-uniformly expanding maps M Stadlbauer, S Suzuki, P Varandas Communications in Mathematical Physics 385 (1), 369-427 , 2021 2021 Citations: 29
On the law of the iterated logarithm for continued fractions with sequentially restricted partial quotients M Stadlbauer, X Zhang Nonlinearity 34 (3), 1389-1407 , 2021 2021 Citations: 5
Ruelle operator for continuous potentials and DLR-Gibbs measures L Cioletti, A Lopes, M Stadlbauer Discrete and continuous dynamical systems 40 (8), 4625-4652 , 2020 2020 Citations: 37
Functions for Group Extensions M Stadlbauer New Trends in One-Dimensional Dynamics: In Honour of Welington de Melo on … , 2019 2019
On conformal measures and harmonic functions for group extensions M Stadlbauer New Trends in One-Dimensional Dynamics: In Honour of Welington de Melo on … , 2019 2019 Citations: 9
Thermodynamic Formalism for Topological Markov Chains on Borel Standard Spaces L Cioletti, EA Silva, M Stadlbauer arXiv preprint arXiv:1810.03757 , 2018 2018 Citations: 19
Coupling methods for random topological Markov chains M Stadlbauer Ergodic Theory and Dynamical Systems 37 (3), 971-994 , 2017 2017 Citations: 30
Spectral properties of the Ruelle operator for product‐type potentials on shift spaces L Cioletti, M Denker, AO Lopes, M Stadlbauer Journal of the London Mathematical Society 95 (2), 684-704 , 2017 2017 Citations: 27
On the Lyapunov spectrum of relative transfer operators M Bessa, M Stadlbauer Stochastics and Dynamics 16 (06), 1650024 , 2016 2016 Citations: 14
Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps BR Kloeckner, AO Lopes, M Stadlbauer Nonlinearity 28 (11), 4117-4137 , 2015 2015 Citations: 23
An extension of Kesten’s criterion for amenability to topological Markov chains M Stadlbauer Advances in mathematics 235, 450-468 , 2013 2013 Citations: 46
On random topological Markov chains with big images and preimages M Stadlbauer Stochastics and Dynamics 10 (01), 77-95 , 2010 2010 Citations: 18
MOST CITED SCHOLAR PUBLICATIONS
An extension of Kesten’s criterion for amenability to topological Markov chains M Stadlbauer Advances in mathematics 235, 450-468 , 2013 2013 Citations: 46
Ruelle operator for continuous potentials and DLR-Gibbs measures L Cioletti, A Lopes, M Stadlbauer Discrete and continuous dynamical systems 40 (8), 4625-4652 , 2020 2020 Citations: 37
Thermodynamic formalism for random countable Markov shifts M Denker, Y Kifer, M Stadlbauer Discrete Contin. Dyn. Syst 22 (1-2), 131-164 , 2008 2008 Citations: 36
Coupling methods for random topological Markov chains M Stadlbauer Ergodic Theory and Dynamical Systems 37 (3), 971-994 , 2017 2017 Citations: 30
Thermodynamic formalism for random non-uniformly expanding maps M Stadlbauer, S Suzuki, P Varandas Communications in Mathematical Physics 385 (1), 369-427 , 2021 2021 Citations: 29
Spectral properties of the Ruelle operator for product‐type potentials on shift spaces L Cioletti, M Denker, AO Lopes, M Stadlbauer Journal of the London Mathematical Society 95 (2), 684-704 , 2017 2017 Citations: 27
Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps BR Kloeckner, AO Lopes, M Stadlbauer Nonlinearity 28 (11), 4117-4137 , 2015 2015 Citations: 23
The return sequence of the Bowen-Series map for punctured surfaces M Stadlbauer Fund. Math 182 (3), 221-240 , 2004 2004 Citations: 23
Thermodynamic Formalism for Topological Markov Chains on Borel Standard Spaces L Cioletti, EA Silva, M Stadlbauer arXiv preprint arXiv:1810.03757 , 2018 2018 Citations: 19
On random topological Markov chains with big images and preimages M Stadlbauer Stochastics and Dynamics 10 (01), 77-95 , 2010 2010 Citations: 18
Lyapunov spectra for KMS states on Cuntz-Krieger algebras M Kesseböhmer, M Stadlbauer, BO Stratmann Mathematische Zeitschrift 256 (4), 871-893 , 2007 2007 Citations: 18
Infinite ergodic theory for Kleinian groups M Stadlbauer, BO Stratmann Ergodic Theory and Dynamical Systems 25 (4), 1305-1323 , 2005 2005 Citations: 16
On the Lyapunov spectrum of relative transfer operators M Bessa, M Stadlbauer Stochastics and Dynamics 16 (06), 1650024 , 2016 2016 Citations: 14
Quenched and annealed equilibrium states for random Ruelle expanding maps and applications M Stadlbauer, P Varandas, X Zhang Ergodic Theory and Dynamical Systems 43 (9), 3150-3192 , 2023 2023 Citations: 11
Minimal distance between random orbits S Gouëzel, J Rousseau, M Stadlbauer Probability Theory and Related Fields 189 (3), 811-847 , 2024 2024 Citations: 9
On conformal measures and harmonic functions for group extensions M Stadlbauer New Trends in One-Dimensional Dynamics: In Honour of Welington de Melo on … , 2019 2019 Citations: 9
Conservativity of random Markov fibred systems M Denker, Y Kifer, M Stadlbauer Ergodic Theory and Dynamical Systems 28 (1), 67-85 , 2008 2008 Citations: 9
Spectral triples and Dixmier trace representations of Gibbs measures: theory and examples L Cioletti, LY Hataishi, AO Lopes, M Stadlbauer Nonlinearity 37 , 2024 2024 Citations: 8
Invariant probabilities for discrete time linear dynamics via thermodynamic formalism AO Lopes, A Messaoudi, M Stadlbauer, V Vargas Nonlinearity 34 (12), 8359-8391 , 2021 2021 Citations: 7
The Bowen-Series map for some free groups M Stadlbauer Mathematisches Institut, Universität Göttingen , 2002 2002 Citations: 6
