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  "body": "\n## Statement\n\nThe following result was obtained by Bestvina and Handel in their 1992 paper where train track maps were originally introduced:\n\nLet *φ* ∈ Out(*F*\u003csub\u003e*k*\u003c/sub\u003e) be irreducible. Then there exists a train track representative of *φ*.\n\nFor a topological representative *f*:*Γ*→*Γ* of an automorphism *φ* of *F*\u003csub\u003e*k*\u003c/sub\u003e the *transition matrix* *M*(*f*) is an *r*x*r* matrix (where *r* is the number of topological edges of *Γ*) where the entry *m*\u003csub\u003e*ij*\u003c/sub\u003e is the number of times the path *f*(*e*\u003csub\u003e*j*\u003c/sub\u003e) passes through the edge *e*\u003csub\u003e*i*\u003c/sub\u003e (in either direction). If *φ* is irreducible, the transition matrix *M*(*f*) is *irreducible* in the sense of the Perron\u0026ndash;Frobenius theorem and it has a unique Perron–Frobenius eigenvalue *λ*(*f*) ≥ 1 which is equal to the spectral radius of *M*(*f*).\n\nOne then defines a number of different *moves* on topological representatives of *φ* that are all seen to either decrease or preserve the Perron–Frobenius eigenvalue of the transition matrix. These moves include: subdividing an edge; valence-one homotopy (getting rid of a degree-one vertex); valence-two homotopy (getting rid of a degree-two vertex); collapsing an invariant forest; and folding. Of these moves the valence-one homotopy always reduced the Perron–Frobenius eigenvalue.\n\nStarting with some topological representative *f* of an irreducible automorphism *φ* one then algorithmically constructs a sequence of topological representatives\n\n*f* = *f*\u003csub\u003e1\u003c/sub\u003e, *f*\u003csub\u003e2\u003c/sub\u003e, *f*\u003csub\u003e3\u003c/sub\u003e,...\n\nof *φ* where *f*\u003csub\u003e*n*\u003c/sub\u003e is obtained from *f*\u003csub\u003e*n*−1\u003c/sub\u003e by several moves, specifically chosen. In this sequence, if *f*\u003csub\u003e*n*\u003c/sub\u003e is not a train track map, then the moves producing *f*\u003csub\u003e*n*+1\u003c/sub\u003e from *f*\u003csub\u003e*n*\u003c/sub\u003e necessarily involve a sequence of folds followed by a valence-one homotopy, so that the Perron–Frobenius eigenvalue of *f*\u003csub\u003e*n*+1\u003c/sub\u003e is strictly smaller than that of *f*\u003csub\u003e*n*\u003c/sub\u003e. The process is arranged in such a way that Perron–Frobenius eigenvalues of the maps *f*\u003csub\u003e*n*\u003c/sub\u003e take values in a discrete substet of $\\mathbb R$. This guarantees that the process terminates in a finite number of steps and the last term *f*\u003csub\u003e*N*\u003c/sub\u003e of the sequence is a train track representative of *φ*.\n\nA consequence (requiring additional arguments) of the above theorem is the following:\n\n- If *φ* ∈ Out(*F*\u003csub\u003e*k*\u003c/sub\u003e) is irreducible then the Perron–Frobenius eigenvalue *λ*(*f*) does not depend on the choice of a train track representative *f* of *φ* but is uniquely determined by *φ* itself and is denoted by *λ*(*φ*). The number *λ*(*φ*) is called the *growth rate* of *φ*.\n- If *φ* ∈ Out(*F*\u003csub\u003e*k*\u003c/sub\u003e) is irreducible and of infinite order then *λ*(*φ*) \u003e 1. Moreover, in this case for every free basis *X* of *F*\u003csub\u003e*k*\u003c/sub\u003e and for most nontrivial values of *w* ∈ *F*\u003csub\u003e*k*\u003c/sub\u003e there exists *C* ≥ 1 such that for all *n* ≥ 1\n\n$$\\frac{1}{C}\\lambda^n(\\phi) \\le ||\\phi^n(w)||_X\\le C \\lambda^n(\\phi),$$\n\nwhere ||*u*||\u003csub\u003e*X*\u003c/sub\u003e is the cyclically reduced length of an element *u* of *F*\u003csub\u003e*k*\u003c/sub\u003e with respect to *X*. The only exceptions occur when *F*\u003csub\u003e*k*\u003c/sub\u003e corresponds to the fundamental group of a compact surface with boundary *S*, and *φ* corresponds to a pseudo-Anosov homeomorphism of *S*, and *w* corresponds to a path going around a component of the boundary of *S*.\n\nUnlike for elements of mapping class groups, for an irreducible *φ* ∈ Out(*F*\u003csub\u003e*k*\u003c/sub\u003e) it is often the case  that\n\n*λ*(*φ*) ≠ *λ*(*φ*\u003csup\u003e−1\u003c/sup\u003e).\n",
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        "locator": "Train track map, section Main result for irreducible automorphisms \u003e Bestvina–Handel theorem for irreducible automorphisms",
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