arXiv subjects

Max Auer

Publications and source records attributed to Max Auer.

6 recordsLinked to original sources

arXiv2026

Quenched limit theorems via mixing of all orders for random dynamical systems

We adapt the notion of mixing of all orders to random dynamical systems and use it to establish quenched limit theorems. Assuming quenched exponential mixing of all orders, we prove both the central limit theorem and the Poisson limit theorem. Compared to existing approaches, this framework offers two main advantages. First, it allows for substantially weaker assumptions on the random constants appearing in mixing estimates: in the exponential mixing regime, logarithmic integrability suffices for the central limit theorem, while in the Poisson case, no integrability condition is required. Second, it applies naturally to invertible systems, where standard methods based on $L^{\infty}$-type estimates or spectral decay are less well suited.

math.DS↗

arXiv2026

Trimmed strong laws and distributional limits for exponentially mixing systems

The Birkhoff Ergodic Theorem establishes pointwise convergence for integrable observables, but for $f\notin L^1$, no normalization yields almost sure convergence. This paper investigates trimmed ergodic sums, where the largest observations are removed, for observables with polynomial tails satisfying $\P(f>t)\,t^{1/\alpha}\to c>0$ as $t\to\infty$, in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when $\alpha\geq 1$, extending known results from the i.i.d.\ case. We further establish distributional limit theorems for both lightly and intermediately trimmed sums in the regime $\alpha>1/2$, showing convergence to an explicitly described non-standard law and to a normal distribution, respectively. The proofs rely on approximating trimmed sums by truncated ergodic sums and exploiting the system's exponential mixing properties.

math.DS↗

arXiv2025

Trimmed ergodic sums for non-integrable functions with power singularities over irrational rotations

Studying Birkhoff sums of non-integrable functions involves the challenge of large observations depending on the sampled orbit, which prevents pointwise limit theorems. To address this issue, the largest observations are removed, this process is commonly known as trimming. While this method is well studied for independent identically distributed sequences and systems with strong mixing behaviour, this paper focuses on irrational rotations of $\mathbb{T}$. In this setting we establish trimmed weak and strong laws for the functions $\frac{1}{x}$ and $\frac{1}{x^{\beta}}$ with $\beta>1$, providing explicit conditions on the rotation angle.

math.DS↗

arXiv2024

Weak mixing and sparse equidistribution

The celebrated Birkhoff Ergodic Theorem asserts that, for an ergodic map, orbits of almost every point equidistributes when sampled at integer times. This result was generalized by Bourgain to many natural sparse subsets of the integers. On the other hand, the behaviour of orbits of \textbf{all} points in a dynamical system is much less understood, especially for sparse subsets of the integers. We generalize a method introduced by A. Venkatesh to tackle this problem in two directions, general $\mathbb{R}^d$ actions instead of flows, and weak mixing, rather than mixing, actions. Along the way, we also establish some basic properties of weak mixing and show weak mixing for the time 1-map of a weak mixing flow.

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arXiv2023

Local limit theorems for hitting times and return times of small sets

We establish abstract local limit theorems for hitting times and return-times of suitable sequences (A_{l}) of asymptotically rare events in ergodic probability preserving dynamical systems, including versions for tuples of consecutive times and positions of the hits. These results are shown to apply in the setup of Gibbs-Markov systems.

math.DS↗

arXiv2023

Poisson Limit Theorems for Systems with Product Structure

We obtain a Poisson Limit for return times to small sets for product systems. Only one factor is required to be hyperbolic while the second factor is only required to satisfy polynomial deviation bounds for ergodic sums. In particular, the second fact can be either elliptic or parabolic. As an application of our main result, several maps of the form Anosov map $\times$ another map are shown to satisfy a Poisson Limit Theorem at typical points, some even at all points. The methods can be extended to certain types of skew products, including $T,T^{-1}$-maps of high rank.

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