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Martin Lustig

Publications and source records attributed to Martin Lustig.

At least 19 recordsLinked to original sources
arXiv2022

Measure transfer and $S$-adic developments for subshifts

Based on previous work of the authors, to any $S$-adic development of a subshift $X$ a "directive sequence" of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given $S$-adic development. The issuing rich picture enables one to deduce results about $X$ with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d \geq 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$.

math.DS↗
arXiv2022

The measure transfer for subshifts induced by a morphism of free monoids

Every non-erasing monoid morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ induces a {\em measure transfer map} $\sigma_X^{\mathcal{M}}: \mathcal{M}(X) \to \mathcal{M}(\sigma(X))$ between the measure cones $\mathcal{M}(X)$ and $\mathcal{M}(\sigma(X))$, associated to any subshift $X \subset \mathcal{A}^{\mathbb{Z}}$ and its image subshift $\sigma(X) \subset \mathcal{B}^{\mathbb{Z}}$ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective \cite{BHL2.8-II}. Furthermore, an efficient technique to compute the value of the transferred measure $\sigma_X^{\mathcal{M}(\mu)}$ on any cylinder $[w]$ (for $w \in \mathcal{B}^*$) is presented. \smallskip \noindent {\bf Theorem:} If a non-erasing morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ is injective on the shift-orbits of some subshift $X \subset \mathcal{A}^\mathbb{Z}$, then $\sigma^{\mathcal{M}_X}$ is injective. \smallskip The assumption on $\sigma$ that it is ``injective on the shift-orbits of $X$'' is strictly weaker than ``recognizable in $X$'', and strictly stronger than ``recognizable for aperiodic points in $X$''. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map $\sigma_X^{\mathcal{M}}$.

math.DS↗
arXiv2022

How do topological entropy and factor complexity behave under monoid morphisms and free group basis changes ?

For any non-erasing free monoid morphism $\sigma: \cal A^* \to \cal B^*$, and for any subshift $X \subset \cal A^\Z$ and its image subshift $Y = \sigma(X) \subset \cal B^\Z$, the associated complexity functions $p_X$ and $p_Y$ are shown to satisfy: there exist constants $c, d, C > 0$ such that $$c \cdot p_X(d \cdot n) \,\, \leq \,\, p_Y(n) \,\, \leq \,\, C \cdot p_X(n)$$ holds for all sufficiently large integers $n \in \N$, provided that $\sigma$ is recognizable in $X$. If $\sigma$ is in addition letter-to-letter, then $p_Y$ belongs to $\Theta(p_X)$ (and conversely). Otherwise, however, there are examples where $p_X$ is not in $\cal O(p_Y)$. It follows that in general the value $h_X$ of the topological entropy of $X$ is not preserved when applying a morphism $\sigma$ to $X$, even if $\sigma$ is recognizable in $X$. As a consequence, there is no meaningful way to define the topological entropy of a current on a free group $F_N$; only the distinction of currents $\mu$ with topological entropy $h_{\tiny\supp(\mu)} = 0$ and $h_{\tiny\supp(\mu)} > 0$ is well defined.

math.DS↗
arXiv2020

Train track maps for graphs of groups

We define train track maps for graphs-of-groups $\cal G$ and exhibit the precise conditions under which the fundamental finiteness properties known for classical train track maps extend to this generalization. These finiteness properties are the crucial tool to control the decrease of illegal turns under iteration of the train track map, and they are a principal ingredient in the answer to basic algorithmic questions about automorphisms induced by such train track maps on $π_1 \cal G$.

math.GR↗
arXiv2020

Invariant measures on finite rank subshifts

In this note we show that for any subshift $X$ of finite $S$-rank every invariant measure $μ$ is determined by its values on finitely many cylinders. Under mild conditions these cylinders are given by the letters of the alphabet in question.

math.DS↗
arXiv2019

Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow Γ$ and vector towers $\overleftarrow v$ on $\overleftarrow Γ$, which allows us to efficiently describe all invariant measures $μ= μ^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $φ: F_N \to F_N$ is determined, when $φ$ is represented by a train track map.

math.DS↗
arXiv2019

Nielsen Equivalence in Fuchsian groups

In this paper we give a complete classification of minimal generating systems in a very general class of Fuchsian groups G. This class includes for example any G which has at least seven non-conjugate cyclic subgroups of order greater than 2. In particular, the well known problematic cases where G has characteristic exponents equal to 2 are not excluded. We classify generating systems up to Nielsen equivalence; this notion is strongly related to Heegaard splittings of 3-manifolds. The results of this paper provide in particular the tools for a rather general extension of previous work of the authors and others, on the isotopy classification of such splittings in Seifert fibered spaces.

math.GT↗
arXiv2019

Tower power for $S$-adics

We explain and restate the results from our recent paper arXiv:1503.08000.v3 in standard language for substitutions and $S$-adic systems in symbolic dynamics. We then produce as rather direct application an $S$-adic system (with finite set of substitutions $S$ on $d$ letters) that is minimal and has $d$ distinct ergodic probability measures. As second application we exhibit a formula that allows an efficient practical computation of the cylinder measure $\mu([w])$, for any word $w \in \cal A^*$ and any invariant measure $\mu$ on the subshift $X_\sigma$ defined by any everywhere growing but not necessarily primitive or irreducible substitution $\sigma: \cal A^* \to \cal A^*$. Several examples are considered in detail, and model computations are presented.

math.DS↗
arXiv2017

Normal form and parabolic dynamics for quadratically growing automorphisms of free groups

We present a normal form for outer automorphisms $ϕ$ of a non-abelian free group $F_N$ which grow quadratically (measured through the maximal growth of conjugacy classes in $F_N$ under iteration of $ϕ$). In analogy to the known normal form for linearly growing automorphisms as efficient Dehn twist, our normal form for $ϕ$ is given in terms of a 2-level Dehn twist on a graph-of-groups $\cal{G}$ with $π_1 {\cal{G}} \cong F_N$, where a conjugacy class of $F_N$ grows at most linearly if and only if it is contained in a vertex group of $\cal{G}$. Our proof is based on earlier work of the second author and on a new cancellation result, which also allows us to show that the dynamics of the induced $ϕ$-action on Outer space $CV_N$ consists entirely of parabolic orbits, with limit points all assembled in the simplex $Δ_{\cal{G}} \subset \partial CV_N$ determined by $\cal{G}$.

math.GR↗
arXiv2017

Long turns, INP's and index for free group automorphisms

The goal of this paper is to introduce a new tool, called {\em long turns}, which is a useful addition to the train track technology for automorphisms of free groups, in that it allows one to control periodic INPs in a train track map and hence the index of the induced automorphism.

math.GR↗
arXiv2017

Index realization for automorphisms of free groups

For any surface $Σ$ of genus $g \geq 1$ and (essentially) any collection of positive integers $i_1, i_2, \ldots, i_\ell$ with $i_1+\cdots +i_\ell = 4g-4$ Masur and Smillie have shown that there exists a pseudo-Anosov homeomorphism $h:Σ\to Σ$ with precisely $\ell$ singularities $S_1, \ldots, S_\ell$ in its stable foliation $\cal L$, such that $\cal L$ has precisely $i_k+2$ separatrices raying out from each $S_k$. In this paper we prove the analogue of this result for automorphisms of a free group $F_N$, where "pseudo-Anosov homeomorphism" is replaced by "fully irreducible automorphism" and the Gauss-Bonnet equality $i_1+\cdots +i_\ell = 4g-4$ is replaced by the index inequality $i_1+\cdots +i_\ell \leq 2N-2$ from Gaboriau, Jaeger, Levitt and Lustig.

math.GR↗
arXiv2016

Perron-Frobenius theory and frequency convergence for reducible substitutions

We prove a general version of the classical Perron-Frobenius convergence property for reducible matrices. We then apply this result to reducible substitutions and use it to produce limit frequencies for factors and hence invariant measures on the associated subshift. The analogous results are well known for primitive substitutions and have found many applications, but for reducible substitutions the tools provided here were so far missing from the theory.

math.DS↗
arXiv2014

Cannon-Thurston fibers for iwip automorphisms of $F_N$

For any atoroidal iwip $ϕ\in Out(F_N)$ the mapping torus group $G_ϕ=F_N\rtimes_ϕ e$ is hyperbolic, and the embedding $ι: F_N \overset{\lhd}{\longrightarrow} G_ϕ$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat ι: \partial F_N \to \partial G_ϕ$. We prove that for any $ϕ$ as above, the map $\hat ι$ is finite-to-one and that the preimage of every point of $\partial G_ϕ$ has cardinality $\le 2N$. We also prove that every point $S\in \partial G_ϕ$ with $\ge 3$ preimages in $\partial F_N$ has the form $(wt^m)^\infty$ where $w\in F_N, m\ne 0$, and that there are at most $4N-5$ distinct $F_N$-orbits of such {\em singular} points in $\partial G_ϕ$ (for the translation action of $F_N$ on $\partial G_ϕ$). By contrast, we show that for $k=1,2$ there are uncountably many points $S\in \partial G_ϕ$ (and thus uncountably many $F_N$-orbits of such $S$) with exactly $k$ preimages in $\partial F_N$.

math.GR↗
arXiv2013

Invariant laminations for irreducible automorphisms of free groups

For every atoroidal iwip automorphism $ϕ$ of $F_N$ (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree $T_+(ϕ)$ is obtained as "diagonal closure" of the support of the backward limit current $μ_-(ϕ)$. This diagonal closure is obtained through a finite procedure in analogy to adding diagonal leaves from the complementary components to the stable lamination of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of $T_+(ϕ)$, in terms of Bestvina-Feighn-Handel's "stable lamination" associated to $ϕ$.

math.GR↗
arXiv2013

Tree-irreducible automorphisms of free groups

We introduce a new class of automorphisms $φ$ of the non-abelian free group $F_N$ of finite rank $N \geq 2$ which contains all iwips (= fully irreducible automorphisms), but also any automorphism induced by a pseudo-Anosov homeomorphism of a surface with arbitrary many boundary components. More generally, there may be subgroups of $F_N$ of rank $\geq 2$ on which $φ$ restricts to the identity. We prove some basic facts about such {\em tree-irreducible} automorphisms, and show that, together with Dehn twist automorphisms, they are the natural basic building blocks from which any automorphism of $\FN$ can be constructed in a train track set-up. We then show: {\bf Theorem:} {\it Every tree-irreducible automorphism of $F_N$ has induced North-South dynamics on the Thurston compactification $\bar{\rm CV}_N$ of Outer space.} Finally, we define a "blow-up" construction on the vertices of a train track map, which, starting from iwips, produces tree-irreducible automorphisms which in general are not iwip.

math.GR↗
arXiv2013

Dual automorphisms of free groups

For any choice of a basis $\cal A$ the free group $F_N$ of finite rank $N \geq 2$ can be canonically identified with the set $F(\cal A)$ of reduced words in $\cal A\cup \cal A^{-1}$. However, such a word $w \in F(\cal A)$ admits a second interpretation, namely as cylinder $C^1_w \subset \partial F_N$. The subset of $\partial F_N$ defined by $C^1_w$ depends not only on the element of $F_N$ given by the word $w$, but also on the chosen basis $\cal A$. In particular one has in general, for $Φ\in \Aut(F_N)$: $$Φ(C^1_w) \neq C^1_{Φ(w)}$$ Indeed, the image of a cylinder under an automorphism $Φ\in \Aut(F_N)$ is in general not a cylinder, but a finite union of cylinders: $$Φ(C^1_w)=C^{1}_U := \bigcup_{u_i \in U} C^1_{u_i}$$ In his thesis the first author has given an efficient algorithm and a formula how to determine such a (uniquely determined) finite {\em reduced} set $U = U(w) \subset F_N$. We use those to define the dual automorphism $Φ_{\cal A}^*$ by setting $Φ_{\cal A}^*(w) = U(w)$. \smallskip \noindent {\bf Theorem:} {\it For any $Φ\in \Aut(F_N)$ there are at most 2N distinct finite subsets $U_i \subset F_N$ such that for any $w = y_1 ... y_r \in F_A$ there is one of them, say $U_{i(w)}$, with $$Φ_{\cal A}^*(w) = Φ(w) U_{i(w)}\, ,$$ and $U_{i(w)}$ depends only on the last letter $y_r \in \CA \cup \CA^{-1}$. Furthermore, the seize of each $U_{i}$ is bounded by $2^t$, where $t \geq 0$ is the number of Nielsen automorphisms in any decomposition of $Φ$ as product of basis permutations, basis inversions and elementary Nielsen automorphisms.}

math.GR↗