## Publications

### On Minimal Subshifts of Linear Word Complexity with Slope less than 3/2

##### Darren Creutz and Ronnie Pavlov

##### On Minimal Subshifts of Linear Word Complexity with Slope less than 3/2 pdf

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.

##### ABSTRACT

### Measure-Theoretically Mixing Subshifts of Minimal Word Complexity

##### Darren Creutz

##### Measure-Theoretically Mixing Subshifts of Minimal Word Complexity pdf

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.

##### ABSTRACT

### Low Complexity Subshifts have Discrete Spectrum

##### 2023

##### Darren Creutz and Ronnie Pavlov

##### Forum of Mathematics, Sigma

##### Low Complexity Subshifts have Discrete Spectrum pdf

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.

##### ABSTRACT

### Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts

##### 2023

##### Darren Creutz

##### Ergodic Theory and Dynamical Systems

##### Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts pdf

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.

##### ABSTRACT

### Stabilizers of Stationary Actions of Lattices in Semisimple Groups

##### 2023

##### Darren Creutz

##### Groups, Geometry and Dynamics

##### Stabilizers of Stationary Actions of Lattices in Semisimple Groups pdf

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.

##### ABSTRACT

### Measure-Theoretically Mixing Subshifts with Low Complexity

##### 2022

##### Darren Creutz, Ronnie Pavlov and Shaun Rodock

##### Ergodic Theory and Dynamical Systems (https://www.doi.org/10.1017/etds.2022.42)

##### Measure-Theoretically Mixing Subshifts with Low Complexity pdf

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.

##### ABSTRACT

### Character Rigidity for Lattices and Commensurators

##### 2022

##### Darren Creutz and Jesse Peterson

##### American Journal of Mathematics

##### Character Rigidity for Lattices and Commensurators pdf

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.

##### ABSTRACT

### Harmonic Functions on Compactly Generated Groups

##### 2022

##### Darren Creutz

cite##### International Journal of Mathematics (Vol. 33, No. 4)

##### Harmonic Functions on Compactly Generated Groups pdf

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).

##### ABSTRACT

### Mixing on Stochastic Staircase Transformations

##### 2021

##### Darren Creutz

cite##### Studia Mathematica (Vol. 257 p. 121–153)

##### Mixing on Stochastic Staircase Transformations pdf

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.

##### ABSTRACT

### Stabilizers of Actions of Lattices in Products of Groups

##### 2017

##### Darren Creutz

cite##### Ergodic Theory and Dynamical Systems (Vol. 37 p. 1133–1186)

##### Stabilizers of Actions of Lattices in Products of Groups pdf

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.

##### ABSTRACT

### Stabilizers of Ergodic Actions of Lattices and Commensurators

##### 2017

##### Darren Creutz and Jesse Peterson

cite##### Transactions of the AMS (Vol. 369 p. 4119–4166)

##### Stabilizers of Ergodic Actions of Lattices and Commensurators pdf

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.

##### ABSTRACT

### Contractive Spaces and Relatively Contractive Maps

##### 2016

##### Darren Creutz

cite##### AMS Contemporary Mathematics (Vol. 678 p. 103–132)

##### Contractive Spaces and Relatively Contractive Maps pdf

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.

##### ABSTRACT

### A Normal Subgroup Theorem for Commensurators of Lattices

##### 2014

##### Darren Creutz and Yehuda Shalom

cite##### Groups, Geometry and Dynamics (Vol. 8 p. 789–810)

##### A Normal Subgroup Theorem for Commensurators of Lattices pdf

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.

##### ABSTRACT

### Mixing on Rank-One Transformations

##### 2010

##### Darren Creutz and Cesar Silva

cite##### Studia Mathematica (Vol. 199 p. 43–72)

##### Mixing on Rank-One Transformations pdf

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.

##### ABSTRACT

### Mixing on a Class of Rank-One Transformations

##### 2004

##### Darren Creutz and Cesar Silva

cite##### Ergodic Theory and Dynamical Systems (Vol. 24 p. 407–440)

##### Mixing on a Class of Rank-One Transformations pdf

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

##### ABSTRACT

### Ergodic Theory of Group Actions

##### Darren Creutz

cite##### Lecture Notes

##### Ergodic Theory of Group Actions pdf

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.

##### ABSTRACT

### Commensurated Subgroups and the Dynamics of Group Actions on Quasi-Invariant Measure Spaces

##### 2011

##### Darren Creutz

cite##### Doctoral Dissertation

##### Commensurated Subgroups and the Dynamics of Group Actions on Quasi-Invariant Measure Spaces pdf

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.

##### ABSTRACT

### Rank-One Mixing and Dynamical Averaging

##### 2001

##### Darren Creutz

cite##### Honors Thesis

##### Rank-One Mixing and Dynamical Averaging pdf

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.

##### ABSTRACT

### Talks

- Adelic subshifts Pennsylvania State University 29 Mar 2024
- Adelic structure of low complexity subshifts Vanderbilt University 22 Mar 2024
- Mixing properties of subshifts and word complexity cutoffs University of Denver Colloquium 8 Mar 2024
- Actions of Lattices in Semisimple Lie Groups University of Denver 1 Mar 2024
- Word Complexity Cutoffs for Mixing Properties of Subshifts University of California: San Diego 8 Feb 2024
- Word Complexity Cutoffs for Mixing Properties of Subshifts Pennsylvania State University 12 Dec 2023
- Actions of Lattices in Semisimple Lie Groups Pennsylvania State University 8 Dec 2023
- Subshift Word Complexity and Mixing Properties University of Denver 14 Nov 2023
- Workshop on Low Complexity Dynamical Systems (co-organizer) University of Maryland 2-6 Oct 2023
- Low Complexity Systems U.S. Naval Academy 18 Apr 2023
- Word Complexity Cutoffs for Mixing Properties of Subshifts University of Maryland 13 Oct 2022
- Word Complexity Cutoffs for Mixing Properties of Subshifts U.S. Naval Academy 3 Oct 2022
- Word Complexity of Rank-One Subshifts University of Denver 26 May 2022
- Low Complexity Subshifts admitting Mixing Measures AMS Special Session on Zero-Dimensional Dynamics, University of Denver (Virtual) 14 May 2022
- Mixing Subshifts with Low Word Complexity Univeristy of Virginia 11 May 2022
- Low Complexity Subshifts U.S. Naval Academy 11 Apr 2022
- Continuous Actions of Lattices (in Higher-Rank Semisimple Groups) are Free U.S. Naval Academy 14 Oct 2021
- Continuous Actions of Lattices (in Higher-Rank Semisimple Groups) are Free University of Maryland 30 Sep 2021
- Complexity from Simplicity: Dimensions, Fractals and Chaos U.S. Naval Academy (online) 23 Feb 2021
- Ergodic Theory: The Mathematics of Experiment, Probability and Time Colloquium, U.S. Naval Academy 13 Nov 2019
- Mixing of Stochastic Staircase Transformations U.S. Naval Academy 21 Oct 2019
- Fractals and Dimension Annapolis Civitan Club 15 Aug 2019
- Mathematical Existence: A Philosophical Discussion U.S. Naval Academy 16 Apr 2019
- Harmonicity and energy of cocycles: (T) without ultraproducts nor conditionally negative type University of Virginia 5 Apr 2019
- Rigidity for actions and characters of lattices (in products of Lie groups) Colloquium, University of Virginia 4 Apr 2019
- Relative Entropy and an Entropy Gap for Some Groups Without (T) Purdue University 15 Feb 2019
- How discontinuous can an integrable function be? U.S. Naval Academy 22 Jan 2019
- Completeness, Incompleteness, Consistency and Independence: GĂ¶del's revolutionizing of logic U.S. Naval Academy 11 Oct 2018
- Amenability of Stationary Actions of Lattices Vanderbilt University 6 Apr 2018
- Amenability of Stationary Actions of Lattices University of Virginia 30 Mar 2018
- A (potential) topological/C*-algebra approach to rigidity for lattices (joint w/ M. Kalantar) U.S. Naval Academy 5 Mar 2018
- Stationary Actions of Lattices Brazos Analysis Seminar, University of Houston 4 Nov 2017
- Relativized Furstenberg Entropy and an Information Theory of Joinings U.S. Naval Academy 16 Oct 2017
- Set Theory Past ZFC: Large Cardinals, Forcing, and Independence U.S. Naval Academy 3 Apr 2017
- The Normal Subgroup Theorem for Lattices in Products (Property (T)) U.S. Naval Academy 6 Feb 2017
- The Normal Subgroup Theorem for Lattices in Products (Amenability) U.S. Naval Academy 23 Jan 2017
- Random Walks and Harmonic Functions on Groups U.S. Naval Academy 7 Nov 2016
- Ergodic Actions of Lattices in Higher-Rank Semisimple Groups University of Maryland 6 Oct 2016
- Mixing and Rank-One Transformations U.S. Naval Academy 3 Oct 2016
- Character Rigidity for Lattices in Lie Groups U.S. Naval Academy 26 Aug 2016
- Rigidty Theory of Lattices in Semisimple Groups U.S. Naval Academy 3 Feb 2016
- The Information Theory of Joinings Vanderbilt University 22 Jan 2016
- Harmonic Maps on Groups and Property (T) Noncommutative Geometry and Operator Algebras Spring Institute 6 May 2015
- Co-Organizer, Special Session: Classification Problems in Operator Algebras AMS Joint Mathematics Meetings 11 Jan 2015
- Operator-Algebraic Superrigidity for Lattices AMS Special Session on Classification Problems in Operator Algebras, Baltimore Maryland 15 Jan 2014
- Operator Algebraic Superrigidity for Lattices and Commensurators Northwestern University 3 Dec 2013
- Rigidity for Characters on Lattices and Commensurators Vanderbilt University 30 Oct 2013
- Character Rigidity for Lattices and Commensurators Vanderbilt University 27 Sep 2013
- Stabilizers of Ergodic Actions of Product Groups and Lattices in Products Shanks Workshop on von Neumann Algebras and Ergodic Theory, Vanderbilt University 28 Apr 2013
- Stabilizers of Actions of Groups and Invariant Random Subgroups Vanderbilt University 26 Apr 2013
- Stabilizers of Actions of Product Groups and Lattices in Product Groups Vanderbilt University 5 Apr 2013
- Mixing on Rank-One Transformations Vanderbilt University 25 Jan 2013
- Stabilizers of Ergodic Actions of Lattices and Commensurators University of California: San Diego 16 Nov 2012
- Poisson Boundaries, Harmonic Functions and Random Walks on Groups Vanderbilt University 9 Nov & 5 Dec 2012
- Stabilizers of Ergodic Actions of Lattices and Commensurators Vanderbilt University 19 Sep 2012
- Stabilizers of Ergodic Actions of Lattices and Commensurators Williams College Ergodic Theory Conference 28 Jul 2012
- Stabilizers of Ergodic Actions of Lattices and Commensurators UCLA Workshop on von Neumann Algebras and Ergodic Theory 26 May 2012
- Stabilizers for Ergodic Actions of Commensurators Vanderbilt University 6 Apr 2012
- The Property (T) “Half” of the Margulis-Zimmer Conjecture Vanderbilt University 29 Feb 2012
- Property (T) for Certain Totally Disconnected Groups Related to a Conjecture of Margulis and Zimmer Vanderbilt University 17 Feb 2012
- SAT Actions and Rigidity of Lattices Vanderbilt University 30 Nov 2011
- Normal Subgroups of Commensurators of Lattices Vanderbilt University 9 Nov 2011
- Normal Subgroups of Commensurators and Rigidity of SAT Actions Vanderbilt University 26 Aug & 2 & 9 Sep 2011
- Dynamics of SAT Actions CalTech 2 May 2011
- Normal Subgroups of Commensurators and Rigidity of SAT Actions University of California: Los Angeles 6 Apr & 13 Apr 2011
- Normal Subgroups and Rigidity for Commensurators Vanderbilt University 28 Feb 2011
- Quasi-Invariant Group Actions University of California: Los Angeles 18 Feb 2011
- Normal Subgroups of Commensurators and SAT Actions CalTech 20 Jan 2011
- A Normal Subgroup Theorem for Commensurators Yale University 15 Nov 2010
- Mixing, Random Sequences and Rank-One Transformations Northwestern University 9 Nov 2010
- Normal Subgroup and Factor Theorems for Commensurators University of Illinois: Chicago 8 Nov 2010
- A Normal Subgroup Theorem for Commensurators of Lattices AMS Western Meeting 9 Oct 2010
- Superstability and Finite-Time Extinction for Semigroups University of California: Los Angeles 27 Apr 2010
- Poisson Boundaries and Their Applications University of California: Los Angeles Jan 2009
- Rank-One Actions, Mixing and Singular Spectra University of California: Los Angeles Mar 2007