Non-Uniquely Ergodic Interval Exchange Transformations

• Non-uniquely ergodic interval exchange transformations are piecewise isometries defined by permutations and length vectors that admit multiple invariant probability measures, often enriched by flips.
• Explicit constructions, such as Kobzev’s triple-measure FIET on eight intervals, use advanced Rauzy induction techniques to delineate the separation and singularity of ergodic measures.
• Recent studies provide sharp bounds on the number of ergodic measures and reveal intricate fractal and parameter-space geometry, offering new insights into the dynamics of FIETs.

A non-uniquely ergodic interval exchange transformation (IET) is a piecewise isometry on an interval, defined by a permutation and a length vector partitioning the interval into subintervals, such that the dynamical system admits more than one invariant ergodic probability measure. When orientation reversals ("flips") are included in the definition (FIET), the complexity of the invariant measure structure increases substantially. The phenomenon of non-unique ergodicity in IETs—particularly those with flips—encapsulates fundamental questions in ergodic theory, combinatorics, and low-dimensional dynamical systems, with explicit examples, sharp bounds, and parameter-space structure now illuminated by recent constructions and dimension-theoretic results.

1. Definitions and Fundamental Principles

Let $I=[0,L)$ be an interval, partitioned into %%%%1%%%% subintervals $[x_{i-1},x_i)$ where $x_0=0$, $x_i=x_{i-1}+\lambda_i$, and $$\lambda=(\lambda_1,\dots,\lambda_n)\in\mathbb{R}_{>0}^n$$, $\sum_{i=1}^n \lambda_i=L$. An IET $T$ is defined by a pair of permutations $(\pi_t,\pi_b)$ on $\{1,\dots,n\}$ (top and bottom orderings) and, in the flipped case, a flip-set $F\subset\{1,\dots,n\}$, acting by a translation or orientation reversal:

• If $i\notin F$: $T$ maps $I_i$ to $I^{\pi}_i$ by translation.
• If $i\in F$: $T$ maps $I_i$ to $I^{\pi}_i$ by $x\mapsto \mathrm{const} - x$.

The system is minimal if every orbit is dense; it is non-uniquely ergodic if there exist at least two distinct ergodic invariant probability measures, typically mutually singular. In the classical (no-flip) case, almost every IET is uniquely ergodic by the Masur–Veech theorem, with the maximal possible number of ergodic probability measures for a minimal $n$-IET given by $\lfloor n/2\rfloor$ (Fickenscher, 2012, Kobzev, 28 Dec 2025).

2. Explicit Construction of Non-Uniquely Ergodic FIETs

The first explicit example of a FIET with three distinct invariant ergodic measures is realized on eight intervals, with specific permutations and a flip-set $F=\{2,3,4,5,6,7\}$ (Kobzev, 17 Dec 2025). The Rauzy induction method is generalized for flips: at each step, one compares the rightmost intervals in both orderings and constructs elementary transition matrices $E_{W,L}$ in $SL_8(\mathbb{Z})$. Kobzev's construction uses a carefully engineered Rauzy path, parameterized by five large integers $(p_1,\dots,p_5)$, producing a return matrix $\Theta=\Theta_{\gamma_p}^3$ whose normalized products define limit vectors $x^{(2)}, x^{(5)}, x^{(7)}$, giving rise to measures $\lambda_2, \lambda_5, \lambda_7$.

The separation of these measures is demonstrated by combinatorial mass estimates: for example, $\lambda_7(I_7)>1/7$, $\lambda_7(I_2)<1/p_1$, etc. This validates the non-uniqueness and mutual singularity of the ergodic measures.

Prior constructions, notably Bas–López (Bas et al., 2020), established only two ergodic measures for (10,$k$)-FIETs, employing infinite Rauzy paths and explicit matrix block structures to exhibit two limiting invariant directions in parameter space (cone of measures), tying measure multiplicity tightly to specific combinatorial motifs and growth rates of matrix products. These constructions set precedent for higher-dimensional analogues.

3. Bounds on the Number of Ergodic Measures

For classical IETs, Veech’s general bound (Fickenscher, 2012, Kobzev, 28 Dec 2025):

$$\#(\text{ergodic measures}) \leq g(\pi) = \frac{1}{2}\operatorname{rank}\Omega_{\pi}$$

where $\Omega_{\pi}$ is a skew-symmetric "intersection matrix" attached to the permutation, and $g(\pi)$ the genus of any suspension. This bound is sharp, with explicit block constructions of self-inverse, Lagrangian permutations yielding exactly $g$ extremal rays in the invariant measure cone.

For FIETs, recent results imply a conjectural upper bound for the number of ergodic measures at $\lfloor n/2\rfloor-1$ for $n$ intervals (Kobzev, 17 Dec 2025), with Kobzev’s triple-measure example at $n=8$ suggesting the conjecture’s tightness.

Fickenscher’s cycle-based approach in (Kobzev, 28 Dec 2025) constructs for every even $n$ an IET with exactly $\lfloor n/2 \rfloor$ ergodic measures, via iterated Rauzy cycles and explicit parameter recursion, saturating Katok’s upper bound via combinatorial domination in matrix product sequences.

4. Rauzy Induction and Measure Construction

Measure construction in FIETs and IETs follows by infinite product of normalized cocycle matrices along carefully selected Rauzy paths:

$$x^{(j)} = \lim_{m\to\infty} \left( \overline{\Theta}_1 \cdots \overline{\Theta}_m \right) e_j$$

where $\overline{\Theta} = \Theta / \|\Theta\|_1$, and $e_j$ denoting basis vectors. The existence and singularity of the measures rely on the doubly-exponential growth (or separation) of certain coordinates under these products, as parameter sequences ($p_i$, $a_m$, $c_m$) are chosen to amplify specific columns.

In flipped systems, the induction strategy ensures the permutation and flip-set recur cyclically, with each path segment engineered to favor mass concentration on predetermined subintervals. The ergodic measures are the limits of empirical distributions started in these intervals or, equivalently, extremal rays of the limiting positive cones under matrix iteration (Kobzev, 17 Dec 2025, Kobzev, 28 Dec 2025).

5. Hausdorff Dimension and Parameter-Space Geometry

Dimension-theoretic analysis reveals linear growth of the Hausdorff dimension $d$ of both the set of minimal FIETs and the non-uniquely ergodic subset, i.e.

• For minimal $n$-FIETs: $n-2 \le d < n-1$.
• For non-uniquely ergodic: $\left\lfloor (n-1)/2 \right\rfloor - 1 \le d$ (Skripchenko et al., 2015).

This "largeness" in the measure-theoretic sense sharply contrasts the Lebesgue-a.e. non-minimal and periodic behavior prevalent in FIET parameter spaces. The fractal structure of invariant measure sets is reflected in the full Hausdorff dimension of constructed measures: under certain parameter growth laws, $\dim_H(\mu_i,d_{\mu_j})=1$ (Kobzev, 28 Dec 2025, Chaika, 2011, Chaika et al., 2014).

Dimension estimates exploit distortion bounds and fast decay properties of induced Markov maps, tracking matrix cocycle growth and return-time scaling; key lemmas (Kerckhoff, distortion, fast-tail) establish exponential tail properties required for upper bounds and repeller structure (Skripchenko et al., 2015).

6. Generalizations, Open Problems, and Future Directions

Current constructions resolve the existence of minimal non-uniquely ergodic FIETs for specific interval and flip patterns, attaining the maximal measure count in classical settings (Kobzev, 28 Dec 2025). Open problems include:

• Sharpness of lower bounds and global upper bounds for the Hausdorff dimension of non-uniquely ergodic sets.
• Classification of measure structure for generalized permutations and partially flipped IETs.
• Determination of the precise maximal measure count for arbitrary $n$-FIETs, with conjectural bound $\lfloor n/2\rfloor-1$ remaining open but plausible by analogy to classical (no-flip) cases.
• Multifractal spectrum analysis and extensions to more general dynamical regimes.

The field also explores the relation between combinatorial motifs in Rauzy diagrams and singularity/fractality of invariant measures, with construction techniques applicable to symbolic substitutions, Arnoux–Rauzy systems, and even circle exchange transformations (Arnoux et al., 2019, Kobzev, 17 Dec 2025). The border of unique ergodicity is mapped to combinatorial, dynamical, and geometric constraints, with new explicit examples extending the taxonomy of interval exchanges beyond the classical paradigm (Ferenczi et al., 2021).

7. Comparative Table: Classical vs. Flipped Non-Uniquely Ergodic IETs

Property	Classical IETs	Flipped IETs (FIETs)
Max. ergodic measures	$\lfloor n/2\rfloor$	Conjectural $\lfloor n/2\rfloor-1$
Typical ergodicity	Almost always unique	Typical: not minimal, not uniquely ergodic (Skripchenko et al., 2015)
Explicit multi-measure examples	Block/Keane/Fickenscher (Fickenscher, 2012, Kobzev, 28 Dec 2025)	Triple-measure FIET (Kobzev, 17 Dec 2025), (10,$k$)-FIET (Bas et al., 2020)
Hausdorff dimension growth	Linear: $n-1$ (minimal), $n-2$ (NUE)	Linear: $n-2$ (minimal), $\lfloor (n-1)/2\rfloor - 1$ (NUE)
Measure separation	By matrix product rays, genus	By parameter growth in flips, Rauzy paths

The interplay between combinatorics, geometry, and parameter selection continues to drive advances in the explicit construction and classification of non-uniquely ergodic interval exchange dynamics, with flipped systems enriching the landscape of measure-theoretic complexity and fractal geometry in measurable dynamics.