A train track directed random walk on
Abstract.
Several known results, by Rivin, CalegariMaher and Sisto, show that an element , obtained after steps of a simple random walk on , is fully irreducible with probability tending to 1 as . In this paper we construct a natural “train track directed” random walk on (where ). We show that, for the element , obtained after steps of this random walk, with asymptotically positive probability the element has the following properties: is ageometric fully irreducible, which admits a train track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that has “rotationless index” (so that the geometric index of the attracting tree of is ), has index list and the ideal Whitehead graph being the complete graph on vertices, and that the axis bundle of in the Outer space consists of a single axis.
2010 Mathematics Subject Classification:
Primary 20F65, Secondary 57M1. Introduction
For an integer , an element is called fully irreducible (sometimes also referred to as irreducible with irreducible powers) if there is no such that preserves the conjugacy class of a proper free factor of . A fully irreducible is called geometric if there exists a compact connected surface with one boundary component such that and such that is induced by a pseudoAnosov homeomorphism of ; fully irreducibles that are not geometric are called nongeometric. Bestvina and Handel proved [BH92] that a fully irreducible is nongeometric if and only if is atoroidal, that is, no positive power of preserves the conjugacy class of a nontrivial element of . It was later shown, as a consequence of the BestvinaFeighn Combination Theorem [BF92], that a fully irreducible is nongeometric if and only if the mapping torus group is wordhyperbolic. For this reason nongeometric fully irreducibles are also called hyperbolic. See Section 2.8 below for more details.
Fully irreducible elements of provide a free group analog of pseudoAnosov elements of the mapping class group of a closed hyperbolic surface . Fully irreducibles play a key role in the study of algebraic, geometric, and dynamical properties of . In particular, every fully irreducible admits a train track representative (see Section 2.5 below for precise definitions), and this fact was, in a sense, the starting point in the development of train track and relative train track theory for free group automorphisms. In the structure theory of subgroups of , subgroups containing fully irreducible elements provide basic building blocks of the theory. For example, the Tits Alternative for , established in full generality in [BFH00, BFH05], was first proved in [BFH97] for subgroups of containing a fully irreducible element. A result of Handel and Mosher [HM09], with a recent different proof by Horbez [Hor14b], shows that if is a finitely generated subgroup, then either contains a fully irreducible element or contains a subgroup of finite index in such that preserves the conjugacy class of a proper free factor of . Also, fully irreducible elements are known to have particularly nice properties for the natural actions of on various spaces. In particular, a fully irreducible element acts with “NorthSouth” dynamics on the compactified Outer space (see [LL03]) and with generalized “NorthSouth” dynamics on the projectivized space of geodesic currents , [Mar95, Uya13, Uya14]. For , the “free factor complex” , endowed with a natural action by isometries, is a free group analog of the curve complex of a finite type surface. It is known that is Gromovhyperbolic, and that acts as a loxodromic isometry of if and only if is fully irreducible [BF14].
There are several known results showing that “random” or “generic” elements of are fully irreducible. The first of these results is due to Rivin [Riv08]. He showed that if is a finite generating set of (where ), then for the simple random walk on with respect to (where ), the probability that is fully irreducible goes to as . Rivin later improved this result to show [Riv10] that, with probability tending to as , the element is in fact a nongeometric fully irreducible. Rivin’s approach was homological: he studied the properties of the matrices in coming from the action of on the abelianization of . From the algebraic properties of the characteristic polynomials of these matrices, Rivin was able to derive conclusions about being a nongeometric fully irreducible with probability tending to as . Rivin applied the same method to show [Riv08] that “random” (in the same sense) elements of mapping class groups are pseudoAnosov.
A different, geometric, approach was then explored by Maher [Mah11] in the context of mapping class groups (using the action of the mapping class group on the Teichmuller space), and later by Calegari and Maher [CM10] in the context of group actions of Gromovhyperbolic spaces. Calegari and Maher considered the following general situation. Let be a finitely generated group acting isometrically on a Gromovhyperbolic space and let be a probability measure on with finite support such that this support generates a nonelementary subgroup of . Then Calegari and Maher proved that, for the random walk on determined by , the probability that, for a random trajectory of this walk, the element acts as a loxodromic isometry of tends to exponentially fast as . They established this fact by showing that there exists an such that, in the above situation, with probability tending to exponentially fast as , the translation length of on is . This result applies to many natural situations, such as the action of the mapping class group (or of its “large” subgroup) on the curve complex, and the action of (or of suitably “large” subgroups of ) on the free factor complex . Since an element of acts loxodromically on if and only if this element is fully irreducible, the result of Calegari and Maher implies the result of Rivin if we take to be a finite generating set of and take to be the uniform probability measure on . Recently Mann constructed [Man14] a new Gromovhyperbolic space (quasiisometric to the main connected component of the “intersection graph” defined in [KL05]), obtained as a quotient of and endowed with a natural isometric action of by isometries. Mann showed [Man14] that acts as a loxodromic isometry of if and only if is a nongeometric fully irreducible. The result of CalegariMaher applies to the action of on and thus implies that, for a finitely supported measure on generating a subgroup containing at least two independent nongeometric fully irreducibles, an element , obtained by a random walk of length defined by , is nongeometric fully irreducible with probability tending to exponentially fast, as . Finally, Sisto [Sis11], using a different geometric approach, introduced the notion of a “weakly contracting element” in a group , and showed that weakly contracting elements of are exactly the fully irreducibles. He showed that for any simple random walk on , the element obtained after steps is weakly contracting (and hence fully irreducible) with probability tending to exponentially fast as .
None of the above results yield more precise structural information about “random” elements of , other than the fact that these elements are (nongeometric) fully irreducibles.
There is a considerably more detailed stratification of the set of nongeometric fully irreducibles in terms of their index, their index list, and their ideal Whitehead graph, which we discuss below. The goal of this paper is to derive such detailed structural information for “random” elements of obtained by a certain natural random walk on .
The index theory for elements of is motivated by surface theory. If is a pseudoAnosov element (where is a closed oriented hyperbolic surface), let be the stable measured foliation for . Then has singularities , where is a prong singularity with . In this case it is known that the “index sum” equals exactly . Thus the index sum is a constant independent of , but the “index list” in . is a nontrivial invariant of the conjugacy class of
The original notion of an index for an element of , introduced in [GJLL98], was formulated in terms of the dynamics of the action on the hyperbolic boundary of . This notion of index, in general, is not invariant under replacing by its positive power. Subsequently, more invariant notions of index were developed using tree technology. We discuss the various notions of index for free group automorphisms in Section 2.9 below.
If (where ) is fully irreducible, there is a naturally associated “attracting tree,” endowed with a natural isometric action of (this tree is similar in spirit to the “dual tree” obtained by lifting the stable measured foliation of a pseudoAnosov element of to the universal cover and then collapsing the leaves). See Section 2.9 for the explanation of the construction of from a train track representative of . If is a nongeometric fully irreducible, the action of on is free but highly nondiscrete (in fact, every orbit is dense in ). However, it is known that every branch point in has finite degree, and that there are only finitely many orbits of branch points in . Thus one can still informally view the quotient as a “graph” and, using a formula for what the Euler characteristic of this graph should be, define the notion of a “geometric index” of , where the summation is taken over orbits of branchpoints in ; see Definition 2.28 below. If is a geometric fully irreducible, the action of on is not free, but there is a natural definition of in this case too. Unlike in the surface case, is not a constant in terms of and does depend on the choice of a fully irreducible . For a fully irreducible , the attracting tree depends only on the conjugacy class of in , and in fact for all . Hence is an invariant of the conjugacy class of in , which is also preserved by taking positive powers of . As a consequence of more general results, it is known that, for a fully irreducible , one has and that, for a geometric fully irreducible , one has . Surprisingly, it turns out that for there exist nongeometric fully irreducibles with [BF94, BF95, GJLL98, Gui05, HM07, JL08]; such are called parageometric. A nongeometric fully irreducible with is said to be ageometric.
As we have seen, for a nongeometric fully irreducible , the geometric index arises from an “index sum” over representatives of orbits of branch points in . The terms of this sum provide an “index list,” which is also an invariant of the conjugacy class of , preserved by taking positive powers. In [HM11], Handel and Mosher formalized this fact by introducing the notion of an index list and of rotationless index (the latter is called “index sum” in [HM11]) for a nongeometric fully irreducible . The most invariant definition of these notions involves looking at the structure of branchpoints of , which also shows that for every non geometric fully irreducible . Handel and Mosher also gave an equivalent description of the index list and rotationless index in terms of a train track representative of . We give this description in Definition 2.33 below.
For a nongeometric fully irreducible , Handel and Mosher also introduced another combinatorial object, called the ideal Whitehead graph of , which encodes further, more detailed, information than the index list in a single finite graph. They also provided an equivalent description of in terms of a train track representative of ; see Definition 2.32 below. For a pseudoAnosov, the component of the ideal Whitehead graph coming from a foliation singularity is a polygon with edges corresponding to the lamination leaf lifts bounding a principal region in the universal cover [NH86]. Since the number of vertices of each polygonal ideal Whitehead graph component is determined by the number of prongs of the singularity, the index list and the ideal Whitehead graph record the same data. In the setting, not only is the ideal Whitehead graph a finer invariant (c.f. [Pfa13a, Pfa13b]), but it provides further information about the behavior of lamination leaves at a singularity. It is again an invariant of the conjugacy class of , also invariant under taking positive powers of . Moreover, while is a more detailed structural invariant than or the index list of , both of these invariants can be “readoff” from .
We will now describe the main result of the present paper. Let and let the free group be equipped with a fixed free basis . We denote by the rose, which is a wedge of directed loopedges, wedged at a single vertex and labelled . Thus we have a natural identification .
An elementary Nielsen automorphism of is an element such that there exist , , with the property that , , and for each . We denote such by . We say that an ordered pair is admissible if either and or and . A sequence (where ) of standard Nielsen automorphisms of is called admissible if, for each , the pair is admissible. A sequence of standard Nielsen automorphisms of is called cyclically admissible if it is admissible and if the pair is also admissible. We denote by the set of all elementary Nielsen automorphisms of (so that is a finite set with exactly elements, see Section 5); we also verify in Lemma 5.1 that for every there are exactly elements such that the pair is admissible. It is wellknown that generates a subgroup of finite index in .
We define a finitestate Markov chain with the state set as follows. For we set the transition probability from to to be if the pair is admissible and otherwise. We show in Lemma 5.3 that this is an irreducible aperiodic finite state Markov chain and that the uniform distribution on is stationary for this chain. We then consider a random process defined by this chain starting with the uniform distribution on . Thus can be viewed as a random walk, where we first choose an element uniformly at random and then, if at step we have chosen , we choose according to the distribution defined above. The sample space of is the set of all sequences of elements of and the random walk defines a probability measure on whose support consists of all infinite admissible sequences of . To each trajectory of we associate a sequence , where .
The random walk can be viewed as an version of the simple nonbacktracking random walk on the free group itself. The reason is the following crucial property of admissible sequences: if is an admissible sequence of elements of , then, for every letter , computing the image by performing letterwise substitutions produces a freely reduced word in . This fact, established in Lemma 3.10 below, implies that for any cyclically admissible sequence , the element admits a train track representative on the rose , and, moreover, this train track map has exactly one nondegenerate illegal turn; see Theorem 3.11. That is why we also think of as a “train track directed” random walk on .
In addition, we show in Theorem 6.5 that for each train track map with exactly one nondegenerate illegal turn with , for some positive power of there exists a cyclically admissible sequence such that , and so that our walk reaches (and, moreover, only depends on ).
Definition 1.1 (Property ).
Let be an integer. We say that has property if all of the following hold:

The outer automorphism is ageometric fully irreducible;

We have (so that ), and has singleelement index list .

There exists a train track representative of such that has no pINPs and such that has exactly one nondegenerate illegal turn.

The ideal Whitehead graph of is the complete graph on vertices.

The axis bundle for in consists of a single axis.
(The terms appearing in this definition that have not yet been defined are explained later in the paper).
Our main result (c.f. Theorem 5.7 below) is:
Theorem A.
Let . For let be the event that for a trajectory of the sequence is cyclically admissible. Also, for let be the event that for a trajectory of the outer automorphism has property .
Then the following hold:

For the conditional probability we have

We have and .

For a.e. trajectory of , there exists an such that for every such that is cyclically admissible, we have that the outer automorphism has property .
We then project the random walk to a random walk on by sending each to its transition matrix in , when is viewed as a graph map . We analyze the spectral properties of this projected walk and show that it has positive first Lyapunov exponent, see Proposition 5.13. We then conclude that for a.e. trajectory , the stretch factor grows exponentially in for any increasing sequence of indices such that is cyclically admissible. See Theorem 5.15 below for the precise statement, and see Section 2.6 for the definition and properties of stretch factor for an element of .
As a consequence, we show that our random walk has positive linear rate of escape with respect to the word metric defined by any finite generating set of (c.f. Theorem 5.17):
Theorem B.
Let and let be a finite generating set of such that . Then there exists a constant such that, for a.e. trajectory of ,
Here for , denotes the distance from to in with respect to the word metric on corresponding to .
Note that our random walk is a “left” random walk on , since with a random trajectory of we associate the sequence (rather than ). We explain in Remark 5.8 how one can convert our random walk into a more traditional “right” random walk on , although after such a conversion the statements of our main results become less natural.
The proof of Theorem A is based on completely different methods from all the previous results about the properties of “random” elements of (see above the discussion of the work of Rivin, CalegariMaher, and Sisto). Instead of using the action of on the free factor complex or on the abelianization of , we analyze the properties of train track representatives of elements obtained by our walk . The main payoff is that, apart from concluding that is fully irreducible, we obtain a great deal of extra detailed structural information about the properties of , where such information does not seem to be obtainable by prior methods. A key tool in establishing that is fully irreducible is the train track criterion of full irreducibility obtained in [Pfa13a] (see Proposition 2.27 below); we also discuss a related criterion obtained in [Kap14] (see Proposition 2.26 below). We substantially rely on ideas and results of [Pfa12, Pfa13a, Pfa13b], although the exposition given in the present paper is almost completely selfcontained.
Finally, we pose several open problems naturally arising from our work:
Question 1.2.
In the context of our Theorem A, for a a.e. trajectory and such that is cyclically admissible, what can be said about the rotationless index, index list, and Ideal Whitehead graph of ?
In [JL08], Jäeger and Lustig, for each , constructed a positive automorphism such that is ageometric fully irreducible with and such that , so that is parageometric. In their construction arises as a rather special composition of positive elementary Nielsen automorphisms, where this composition is cyclically admissible in our sense. However, experimental evidence appears to indicate that for produced by our walk for long “random” cyclically admissible compositions, the absolute value of is much smaller than the maximum value of achieved by parageometrics.
Question 1.3.
Again in the context of Theorem A, is it true that for a.e. trajectory of , projecting this trajectory to the free factor complex as , where is a vertex of (or perhaps as ), gives a sequence that converges to a point of the hyperbolic boundary ?
Note that by the recent work of Horbez [Hor14a] on describing the Poisson boundary of , the answer to the similar question for a simple random walk on is positive. In several personal conversations, Camille Horbez indicated to the second author a plausible approach for getting a positive answer to Question 1.3.
Question 1.4.
Let and let be a finite generating set of . If is a random trajectory of the simple random walk on , what can be said about the properties of , apart from the fact that, with probability tending to as , the automorphism is a nongeometric fully irreducible? In particular, is ageometric? What can be said about , and about the index list and the Ideal Whitehead graph of ?
Question 1.5.
Let be a closed oriented hyperbolic surface. What can be said about the index/singularity list for the stable foliation of a “random” element obtained by a simple random walk of length on ? (Note that by the results of Rivin, Maher, and CalegariMaher, discussed above, we do know that is pseudoAnosov with probability tending to as ).
It would also be interesting to understand the index properties of generic automorphisms (where ) produced by a simple random walk on with respect to some finite generating set of . As noted above, it is already known that in this situation is atoroidal and fully irreducible with probability tending to as . Computer experiments, conducted by us using Thierry Coulbois’ computer package for free group automorphisms^{1}^{1}1The package is available at http://www.cmi.univmrs.fr/~coulbois/traintrack/ appear to indicate that generically both and are ageometric fully irreducible, with a very small value of (in contrast with an almost maximal value in Theorem A). These experiments also appear to indicate that several possible index lists for occur with asymptotically positive probability each, with the singleentry list occurring with the highest probability. However, the maximal values of the length of a simple random walk on (with ), that our experiments were able to handle, were around , and longer experiments are needed to get more conclusive empirical data.
A plausible conjecture here would be that all singularities of the stable foliation of a random are 3prong singularities. Note that a result of Eskin, Mirzakhani, and Rafi [EMR12] shows that for “most” (in a different sense) closed geodesics in the moduli space of , the pseudoAnosov element of corresponding to such a closed geodesic has all singularities of its stable foliation being 3prong.
The first author thanks Terence Tao for supplying a proof of Proposition 5.14 and to Jayadev Athreya, Vadim Kaimanovich, Camille Horbez and Igor Rivin for helpful discussions about Lyapunov exponents and random walks. We are also grateful to Lee Mosher for useful conversations regarding the proof of Theorem 6.5.
2. Preliminaries
2.1. Graphs, paths and graph maps
Definition 2.1 (Graphs).
A graph is a 1dimensional cellcomplex. We call the 0cells of vertices and denote the set of all vertices of by . We refer to open 1cells of as topological edges of and denote the set of all topological edges of by .
Each topological edge is homeomorphic to the open interval and thus, when viewed as a 1manifold, admits two possible orientations. An oriented edge of is a topological edge with a choice of an orientation on it. We denote by the set of all oriented edges of . If is an oriented edge, we denote by the same underlying edge with the opposite orientation. Note that for each we have and ; thus is a fixedpointfree involution.
Since is a cellcomplex, every oriented edge of comes equipped with the orientationpreserving attaching map such that maps homeomorphically to and such that . By convention we choose the attaching maps so that, for each and each , we have . For we call the initial vertex of , denoted , and we call the terminal vertex of , denoted . Thus, by definition, and .
If is a graph and , a direction at in is an edge such that . We denote the set of all directions at in by and call it the link of in . Then the degree of in , denoted or , is the cardinality of the set .
An orientation on a graph is a partition such that for an edge we have if and only if .
Note that both topological edges and oriented edges are, by definition, open subsets of and they don’t contain their endpoints.
Definition 2.2 (Combinatorial and topological paths).
A combinatorial edgepath of length is a sequence such that for and such that for all . We put , , and . Thus is again a combinatorial edgepath of length . For we also view as a combinatorial edgepath of length with and . For a combinatorial edgepath of length we denote .
A combinatorial edgepath is reduced or tight if does not contain subpaths of the form , where .
A topological edgepath in is a continuous map such that either and or and there exists a subdivision and a combinatorial edgepath in such that:
(1) We have for .
(2) We have for and for .
(3) is an orientationpreserving homeomorphism mapping onto .
Sometimes we drop the commas and just write .
Note that, for a topological edgepath , where , the combinatorial edgepath with the above properties is unique; we say that is the combinatorial edgepath associated to ; we also call the associated subdivision for . If and , we say the path is associated to .
Let be a topological edgepath (where ), let be the associated combinatorial edgepath, and let be the corresponding subdivision. We say that a topological edgepath is tame if for every the map is a (necessarily unique) orientationpreserving affine homeomorphism. By convention, if is a topological edgepath with , we also consider to be tame.
A topological path (where ) is defined similarly to as in the definition of a topological edgepath above, except that we no longer require and , but instead allow and . For and , condition (3) in the above definition is relaxed accordingly. For we view any map as a topological path in .
For a topological path with there is still a canonically associated combinatorial edgepath and a canonically associated subdivision .
We define what it means for a topological path to be tame, similarly to the notion of a tame topological edgepath above, by requiring all the maps to be injective affine orientationpreserving maps from to subintervals of . For it is still the case that . However, we now allow for the possibility that with (in the case where rather than ) and that with (in the case where rather than ). Also, if , we consider any map to be a tame path in .
Note that if is a topological path (respectively, tame topological path), then for any the restriction is again a topological path (respectively, tame topological path) in .
Also notice that, if and is a combinatorial edgepath and , then there exists a unique tame topological edgepath with associated combinatorial path and associated subdivision , . By contrast, given and , there exist uncountably many topological edgepaths with associated combinatorial path and associated subdivision , . The distinction between topological edgepaths and tame topological edgepaths is often ignored in the literature, but this distinction is important when considering fixed points and dynamics of graph maps, as we will see later.
Definition 2.3 (Paths).
Let be a graph. By a path in we will mean a tame topological path . A path is trivial if and nontrivial if . A path is tight or reduced if the map is locally injective. Thus a trivial path is always tight, and a nontrivial path is tight if and only if the combinatorial edgepath associated to is reduced.
2.2. Graph maps
Definition 2.4 (Graph maps).
Let and be graphs. A topological graph map is a continuous map such that and such that the restriction of to each edge of is a topological edgepath in . More precisely, this means that for each the map is a topological edgepath in .
A graph map is a topological graph map such that the restriction of to each edge of is a path in (in the sense of Definition 2.3), that is, such that for every the map is a tame topological edgepath in .
Convention 2.5.
By convention, if is a tame topological edgepath with the associated combinatorial edgepath , we will usually suppress the distinction between and . In particular, if is a graph map and , we will usually suppress the distinction between the tame topological path and the associated combinatorial edgepath in . Moreover, in this situation we will often write or even .
Note that our definition implies that if is a topological graph map, then for each edge we have with . It is sometimes useful to allow a topological graph map to send an edge to a vertex (rather than to an edgepath of positive combinatorial length), but we will not need this level of generality in the present paper.
A topological graphmap is said to be expanding if for each edge , we have as .
Remark 2.6.
The distinction between the notions of a graph map and of a topological graph map is important when considering the fixed points and the dynamics of a (topological) graph map . Indeed, suppose that is a topological graph map such that for some edge the combinatorial edgepath associated with is , such that , and such that for some we have . Then there exists a subdivision such that maps homeomorphically and preserving orientation to , for . Denote and denote by the open segment in between and ; so that . Thus, for , maps the open segment homeomorphically and preserving orientation to . Our assumption that with implies that the map maps the subsegment of by an orientation preserving homeomorphism to the interval . The intermediate value theorem then implies that there exists such that the point is fixed by , that it satisfies . However, the orientationpreserving homeomorphism can, in principle, have uncountably many fixed points; e.g. could coincide with the identity map on some nondegenerate subsegment of . Thus, may have uncountably many fixed points in the interval of . On the other hand, if in the above situation is a graph map (so that the path is tame), then maps the subinterval of to the interval by an orientation preserving affine homeomorphism. It then follows that there exists a unique such that .
Thus, if is a finite graph and is an expanding (in the combinatorial sense defined above) topological graphmap, then may have uncountably many fixed points in . By contrast, if is finite and is an expanding graphmap, then has only finitely many fixed points and only countably many periodic points in .
Allowing to be a topological graph map, rather than a graph map, may result in some additional pathologies of the dynamics of under iterations; e.g. an expanding topological graphmap may turn out to act as a “contraction” on a nondegenerate subsegment of an edge of . Restricting our consideration to graph maps in this paper rules out these kinds of pathologies.
For a square matrix with real coefficients we denote by the spectral radius of the matrix , that is, the maximum of where varies over all eigenvalues of .
Definition 2.7 (Transition matrix of a graph map).
Let be a finite graph with topological edges. Choose an orientation and an ordering . Let be a graph map. The transition matrix of is an matrix with nonnegative integer entries, where, for , the entry in the position in is equal to the number of times and appear in the combinatorial edgepath .
We denote , the spectral radius of the matrix .
It is not hard to see for the above definition that if are graphmaps, then . In particular, we have for each integer .
Lemma 2.8.
Let be a finite connected graph and let be such that and is surjective. Then .
Proof.
Let be arbitrary. Since , the path passes through every topological edge of . Since is surjective, it follows that the path also passes through every topological edge of . Hence , as required. ∎
Definition 2.9 (Regular map).
A graph map is regular if for each the combinatorial edgepath is reduced. Note that is reduced if and only if the path is locally injective.
Note that, if is a graph map, then for each , we have that is also a graph map. However, if is a regular graph map, then, in general, the map may fail to be regular for some .
2.3. PerronFrobenius theory
We say that a matrix with real coefficients is nonnegative, denoted , if all coefficients of are . Recall that a nonnegative matrix is called irreducible if for each there exists a such that . It is not hard to check that, in the context of Definition 2.7, the matrix is irreducible if and only if for each there exists a such that the path contains an occurrence of either or of .
For a matrix we write if for all . Note that if , then is irreducible.
Recall that for a square matrix with real coefficients we denote by the spectral radius of the matrix .
A key basic result of PerronFrobenius theory says that if is a irreducible matrix then and, moreover, is an eigenvalue for , called the PerronFrobenius (PF) eigenvalue. Moreover, in this case there exists an eigenvector with such that all coefficients of are . See [Sen06] for background on PerronFrobenius theory.
2.4. Train track maps
Definition 2.10 (Train track map).
Let be a finite connected graph without degree1 or degree2 vertices.
A graphmap is called a train track map if the following hold:

is a homotopy equivalence and

for each the graphmap is regular (that, is, for every and every the edgepath is reduced).
The definition above implies that if is a train track map, then for each is also a train track map.
A train track map is said to be irreducible if its transition matrix is irreducible. A train track map is said to be expanding if for every edge we have as . Thus a train track map is expanding if and only if for each there exist such that .
Definition 2.11 (Derivative map).
Let be a graph map. The derivative map is defined as follows. For an edge with