Idempotents in the Ellis semigroup of Floyd-Auslander systems

We study minimal idempotents $J^{\mathrm{min}}(X)$ in the Ellis semigroup $E(X)$ associated with a Floyd-Auslander system $(X,T)$. We show that $(X,T)$ is non-tame if and only if $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$, which happens exactly when the factor map onto the maximal equicontinuous factor possesses uncountably many non-invertible fibres. This yields an easy-to-check criterion for distinguishing tame from non-tame Floyd-Auslander systems and, more importantly, provides an entire family of regular almost automorphic systems with $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$. Notably, all previously known regular almost automorphic non-tame systems exhibited only a small (i.e. $\leq 2^{\aleph_0}$) set of minimal idempotents. We obtain our result by leveraging an alternative characterisation of (non)-tameness through, what we call, choice domains.

💡 Research Summary

The paper investigates the structure of minimal idempotents in the Ellis semigroup of Floyd‑Auslander systems, a class of topological dynamical systems built by repeatedly subdividing the unit square into thin vertical rectangles. The Ellis semigroup (E(X,T)) is defined as the closure (in the product topology) of the set of forward iterates ({T^{n}:n\in\mathbb N}). While the Ellis semigroup is a powerful invariant—encoding properties such as distality, minimality, and the maximal equicontinuous factor—its concrete description is notoriously difficult, especially for non‑tame systems whose semigroup contains a copy of the Stone–Čech compactification (\beta\mathbb N).

The authors focus on the set (J^{\min}(X)) of minimal idempotents (those idempotents that are minimal with respect to the natural partial order (e\le f\iff ef=e)). They prove a striking equivalence for minimal Floyd‑Auslander systems:

1. The system ((X,T)) is non‑tame (i.e., its Ellis semigroup has cardinality larger than the continuum).
2. The set of minimal idempotents satisfies (|J^{\min}(X)|>2^{\aleph_{0}}).
3. The index set (\Lambda={n\in\mathbb N:|Q(n)|\ge2}) is infinite, where (Q(n)) records the positions at stage (n) where the vertical fibre is non‑degenerate (i.e., a genuine interval rather than a single point).

The implication (3) → (2) is the technical heart of the work. To handle it, the authors introduce a new combinatorial notion called a choice domain. For a binary subshift ((X,\sigma)\subset{0,1}^{\mathbb N}), a subset (T\subset X) is a choice domain if, for any finite collection of points from (T) and any prescribed pattern of 0/1 values on a finite set of time indices, one can find a common time schedule that forces each point to realize its prescribed pattern. They prove Theorem B: a binary subshift is non‑tame if and only if it admits an uncountable choice domain. This result is essentially equivalent to the classical characterization of non‑tameness via infinite independence sets, but the choice‑domain formulation is more constructive and better suited for the skew‑product description of Floyd‑Auslander systems.

The Floyd‑Auslander systems are re‑presented as skew‑products over an odometer ((\Sigma(p_n),1)) with fibre maps chosen from three affine contractions (\lambda_0(x)=\frac{x}{2}), (\lambda_1(x)=x), and (\lambda_2(x)=\frac{x}{2}+\frac12). At each level (n) the set (Q(n)) consists of those symbols that select the identity map (\lambda_1); when (|Q(n)|\ge2) the corresponding vertical fibres become non‑degenerate intervals. The authors show that if (\Lambda) is infinite, one can extract from the system an uncountable choice domain inside a suitable binary subshift derived from the symbolic coding of the odometer. By Theorem B this subshift is non‑tame, and consequently the original Floyd‑Auslander system is non‑tame. Since non‑tameness already forces (|J^{\min}(X)|>2^{\aleph_0}) (Lemma 3.5), the three statements are equivalent.

The paper also places these findings in the broader context of regular almost automorphic systems. Previously known examples of regular almost automorphic non‑tame systems exhibited at most continuum many minimal idempotents. The present work provides the first family where the set of minimal idempotents exceeds the continuum, revealing a new phenomenon: the Ellis semigroup can be “large” not only in cardinality but also in the richness of its minimal idempotent structure.

Finally, the authors discuss how the notion of choice domains extends beyond binary subshifts to general topological dynamical systems, offering a practical alternative to the traditional independence‑set criterion for detecting non‑tameness. This new tool may simplify the analysis of other skew‑product or substitution systems where explicit symbolic representations are cumbersome.

In summary, the paper delivers (i) an exact combinatorial criterion (the infinitude of (\Lambda)) for non‑tameness of Floyd‑Auslander systems, (ii) a proof that this criterion is equivalent to the existence of more than continuum many minimal idempotents in the Ellis semigroup, and (iii) the introduction of choice domains as a versatile method for studying tameness in symbolic dynamics. These contributions deepen our understanding of the interplay between algebraic properties of the Ellis semigroup and the dynamical complexity of almost automorphic systems.