We prove that every infinite minimal subshift with word complexity p(q) satisfying limsup p(q)/q < 3/2 is measure-theoretically isomorphic to its maximal equicontinuous factor; in particular, it has measurably discrete spectrum. Among other applications, this provides a proof of Sarnak's conjecture for all subshifts with limsup p(q)/q < 3/2 (which can be thought of as a much stronger version of zero entropy).

As in [CP23], our main technique is proving that all low-complexity minimal subshifts have a specific type of representation via a sequence { tau_k } of substitutions, usually called an S-adic decomposition. The maximal equicontinuous factor is the product of an odometer with a rotation on an abelian one-dimensional nilmanifold with adelic structure, for which we can give an explicit description in terms of the substitutions tau_k. We also prove that all such odometers and nilmanifolds may appear for minimal subshifts with lim p(q)/q = 1, demonstrating that lower complexity thresholds do not further restrict the possible structure.

We resolve a long-standing open question on the relationship between measure-theoretic dynamical complexity and symbolic complexity by establishing the exact word complexity at which measure-theoretic strong mixing manifests:

For every superlinear f : N , i.e. f(q)/q --> ∞, there exists a subshift admitting a (strongly) mixing of all orders probability measure with word complexity p such that p(q)/f(q) --> 0.

For a subshift with word complexity p which is non-superlinear, i.e. liminf p(q)/q < ∞, every ergodic probability measure is partially rigid.

We prove results about subshifts with linear (word) complexity, meaning that limsup p(n)/n < infty, where for every n, p(n) is the number of n-letter words appearing in sequences in the subshift. Denoting this limsup by C, we show that when C < 4/3, the subshift has discrete spectrum, i.e. is measurably isomorphic to a rotation of a compact abelian group with Haar measure. We also give an example with C = 3/2 which has a weak mixing measure. This partially answers an open question of Ferenczi, who asked whether C = 5/3 was the minimum possible among such subshifts; our results show that the infimum in fact lies in [4/3, 3/2]. All results are consequences of a general S-adic/substitutive structure proved when C < 4/3.

We exhibit subshifts admitting weakly mixing (probability) measures, for arbitrary epsilon > 0, with word complexity p satisfying limsup p(q)/q < 1.5 + epsilon.
For arbitrary f(q) --> ∞, said subshifts can be made to satisfy p(q) < q + f(q) infinitely often.

We establish that every subshift associated to a rank-one transformation (on a probability space) which is not an odometer satisfies limsup p(q) - 1.5q = ∞ and that this is optimal for rank-ones.

Every stationary action of a strongly irreducible lattice or commensurator of such a latiice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely. Consequently, every minimal action of such a lattice on an infinite compact metric space is topologically free.

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any f : N -> N with f(n)/n increasing and Σ 1/f(n) < ∞, that there exists an extremely elevated staircase with word complexity p(n) = o(f(n)). This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure preserving actions, and also answers a question of Connes for such groups.

A compactly generated group is noncompact if and only if it admits a nonconstant harmonic function (for some, equivalently for every, reasonable measure).
This generalizes the known fact that a finitely generated group is infinite if and only if it admits a nonconstant harmonic function (for some, equivalently every, reasonable measure).

We prove mixing on a general class of rank-one transformations containing all known examples of rank-one mixing, including staircase transformations and Ornstein's constructions, and a variety of new constructions.

We prove that any ergodic nonatomic probability-preserving action of an irreducible lattice in a semisimple group, at least one factor being connected and higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is higher-rank.

We also prove a generalization of a result of Bader and Shalom by showing that any probability-preserving action of a product of simple groups, at least one with property (T), which is ergodic for each simple subgroup is either essentially free or essentially transitive.

Our method involves the study of relatively contractive maps and the Howe-Moore property, rather than the relaying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups of independent interest.

We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices.

The above are derived from more general results on groups with the Howe-Moore property and property (T). We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.

We present an exposition of contractive spaces and of relatively contractive maps. Contractive spaces are the natural opposite of measure-preserving actions and relatively contractive maps the natural opposite of relatively measure-preserving maps. These concepts play a central role in the work of the author and J.~Peterson on the rigidity of actions of semisimple groups and their lattices and have also appeared in recent work of various other authors. We present detailed definitions and explore the relationship of these phenomena with other aspects of the ergodic theory of group actions, proving along the way several new results, with an eye towards explaining how contractiveness is intimately connected with rigidity phenomena.

We establish a general normal subgroup theorem for commensurators of lattices in locally compact groups. While the statement is completely elementary, its proof, which rests on the original strategy of Margulis in the case of higher rank lattices, relies heavily on analytic tools pertaining to amenability and Kazhdan's property (T). It is a counterpart to the normal subgroup theorem for irreducible lattices of Bader and the second named author, and may also be used to sharpen that result when one of the ambient factors is totally disconnected.

We prove mixing on rank-one transformations is equivalent to “the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums”. In particular, all staircase transformations (and polynomial staircase transformations) are mixing, answering Adams' question.

We prove that a rank-one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences, in the sense that the mean ergodic theorem holds for a family of what we call dynamical sequences.

The application of our theorem shows that the class of polynomial rank-one transformations, rank-one transformations where the spacers are chosen to be the values of a polynomial with some mild conditions on the polynomials, that have restricted growth are mixing transformations implying in particular Adams’ result on staircase transformations. Another application yields a new proof that Ornstein’s class of rank-one transformations constructed using “random spacers” are almost surely mixing transformations.

Addenda and Errata

Ergodic theory is the subfield of dynamics concerned with actions of groups and semigroups on measure spaces. This text covers the basics of classical ergodic theory and then moves to the more modern and more general setting of group actions on probability spaces.

Classical ergodic theory is concerned with Z-actions (or N-actions) on (completions of) standard Borel spaces, usually referred to as transformations (the single map which generates the action). The most common case, and the one primarily considered here, is when the measure of the entire space is finite (and hence can and will be normalized to be a probability measure).

The more modern material focuses especially on the situation of nonamenable groups, where many of the results from the classical theory are not available. The emphasis is on the aspects of ergodic theory that arise in connection with the rigidity theory of lattices in semisimple groups, particularly the aspects arising in the author's research.

Margulis’ Normal Subgroup and Arithmeticity Theorems classify completely the lattices in higher-rank algebraic groups as being (up to finite index) the simple groups consisting of integer points of algebraic groups. We generalize the Normal Subgroup Theorem to commensurators of lattices in arbitrary locally compact groups and in particular obtain that the commensurator of a uniform tree lattice in a regular tree is simple (up to finite index) if and only if a corresponding completion is. We also make progress on the Margulis-Zimmer Conjecture: a generalization of the Arithmeticity Theorem to commensurated--geometrically normal--subgroups by establishing the property (T) “half” of the conjecture.

In keeping with the methods of Margulis’ work and of rigidity theory in general, we prove new results in the area of the dynamics of group actions with quasi-invariant measures and in the area of unitary representation theory that in turn lead to our structural results on commensurators of lattices in arbitrary groups and commensurated subgroups of lattices in algebraic groups.

The dynamics of group actions with quasi-invariant measures–the study of group actions on measure algebras preserving the ideal of null sets, but not necessarily the measure itself–is the natural setting for dynamics of nonamenable groups (such as those we are concerned with). We develop some foundational results on strongly approximately transitive actions–the dynamical opposite of measure-preserving–and prove a “SAT Factor Theorem” along the lines of Margulis’ original boundary-based factor theorem that plays a key role in the proof of our Normal Subgroup Theorem.

In the realm of unitary representations we prove an existence and uniqueness result for harmonic cocycles–namely, in any reduced cohomology class there exists a unique harmonic representative. Consequences of this include our progress on the Margulis-Zimmer Conjecture.

A rank-one transformation is defined by a sequence of positive integers, the sequence of cuts, and a dynamical sequence of nonnegative integers, the sequence of spacers, that are used to repeatedly cut and stack a single column. Our main result is that rank-one transformations satisfying a condition called restricted growth and such that the spacer sequence is uniformly ergodic with respect to the transformation are mixing transformations.

The restricted growth condition limits the total variation in the spacer sequence and is a generalization of a condition, equivalent to restricted growth for staircase transformations, given by Adams, that is sufficient for staircase transformations to be mixing, while the uniform ergodicity of the spacer sequence is a generalization of the notion of uniform Cesaro transformations used by Adams to show mixing on staircases.

The application of our concepts and results to a class of rank-one transformations, a class we call generalized staircase transformations, yields a variety of rank-one mixing transformations with explicit constructions.