The algebraic entropy of the special linear character automorphisms of a free group on two generators

Abstract

Abstract. In this note, we establish a connection between the dynamical degree, or algebraic entropy of a certain class of polynomial automorphisms of R 3 , and the maximum topological entropy of the action when restricted to compact invariant subvarieties. Indeed, when there is no cancellation of leading terms in the successive iterates of the polynomial automorphism, the two quantities are equal. In general, however, the algebraic entropy overestimates the topological entropy. These polynomial automorphisms arise as extensions of mapping class actions of a punctured torus S on the relative SU(2)-character varieties of S embedded in R 3 . It is known that the topological entropy of these mapping class actions is maximized on the relative character variety comprised of reducible characters (those whose boundary holonomy is 2). Here we calculate the algebraic entropy of the induced polynomial automorphisms on the character varieties and show that it too solely depends on the topology of S.

Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals

“…Aside from the context described here, this map appears in a natural way in problems related to dynamics of mapping classes [84], Fuchsian groups [20], number theory [19], Painlevé sixth equations [31,102], the Ising model for quasicrystals [15,90,174,175], the Fibonacci quantum walk [153,154], among others [7,64,166,176]. See [30] or [17] for an algebraic explanation of this universality. We refer the reader also to [97,155,156] for further reading on the Fibonacci trace map.…”

Points with finite orbits for trace maps

“…Many authors have studied trace maps [1,5,7,16,17,19,22,23,24,26], which give an action of Aut(F 2 ) on R 3 (here F n is a rank n free group) and form an interaction between representation theory and dynamical systems. Goldman and others have also studied this action in the context of character varieties, see [10,11,12,3,4,27]. The case n = 2 is that usually studied.…”

Algebraic entropy and the action of mapping class groups on character varieties

“…The topological entropy of mapping class group actions on character varieties has been calculated by Fried for the case S = S 1,1 and G = SU (2) [6] and by Cantat and Loray for reduced character varieties (these are character varieties where the traces of boundary components are fixed) in the case S = S 0,4 , G = SL 2 (C) [3]. The algebraic entropy was calculated by Brown for the case S = S 1,1 and G = SU (2) and a specific embedding of X [2]. In all of the above cases, the entropy calculated was equal to ρ(f ).…”