Finite and infinite invariant measures for adic transformations

Albert M. Fisher; Marina Talet

Home/Archive/arXiv

arXiv · 2510.13071

Finite and infinite invariant measures for adic transformations

Albert M. Fisher·Marina Talet

View original source ↗Read source document ↗Export metadata ↓

Abstract

We classify the invariant Borel measures for adic transformations, where the alphabets have bounded size and the measure is finite on the path space of some sub-Bratteli diagram. We develop a nonstationary version of the Frobenius normal form for a reducible matrix, present an appropriate nonstationary notion of distinguished eigenvector, and prove a nonstationary Frobenius--Victory theorem. This parallels the approach to the stationary case developed by Bezuglyi, Kwiatkowski, Medynets and Solomyak in arXiv:0812.1088 where they classify the locally finite invariant measures. In later work, they also address the nonstationary case. We extend this in two ways. Firstly, in both the stationary and nonstationary settings, we allow for measures which are locally infinite, motivating this extension with examples. Secondly we give a complete classification, presenting a necessary and sufficient condition for a measure which is finite on some subdiagram to be infinite for the original diagram. As part of our program, we introduce a related object called an adic tower and a construction called the canonical cover of the subdiagram: if the measure is finite on the subdiagram but locally infinite on the original space it will be locally finite on the cover space, though the path spaces for the original diagram and the cover are measure-theoretically isomorphic. Our examples include two new models for the Integer Cantor Set transformation of [Fisher1992], one locally finite and one locally infinite, which lead to a new class of examples related to fractal sets of integers: nested circle rotations, where one rotation is embedded in another. The resulting tower measure inside the original rotation can have finite or infinite total mass as determined by our general criterion.

Explore related subjects

math.DS

Keep this discovery

BibTeX RIS

Albert M. Fisher, Marina Talet. 2025-10-15. Finite and infinite invariant measures for adic transformations. https://arxiv.org/abs/2510.13071

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗

arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗

arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗