A bridge between number theory and topology via dynamics
The complexity of a mapping from a surface to itself can be measured by so-called topological entropy. One of the most outstanding questions in topology in dimension 2 is which numbers do appear as topological entropy of some surface mapping. Conjecture was formulated more than 30 years ago, but the question itself remains wide open. Here, researchers at KAIST in the Department of Mathematical Sciences apply a probabilistic method to test the difficulty of the conjecture. They use a ground-breaking result from the late Maryam Mirzakhani (the first female Fields medalist) and Alex Eskin of the University of Chicago.
The dynamical property of a surface mapping is important in topology/geometry. Through the seminal work of the legendary mathematician William Thurston, it was discovered that the structure of 3-dimensional space obtained by first thickening a surface and gluing the boundary parts via a surface mapping is closely related to the dynamics of this gluing mapping. One way of measuring the dynamics of a surface mapping is determing topological entropy. In case of a surface with a negatively curved metric, the topological entropy of a mapping is the logarithm of the exponential growth rate of the length of a simple loop under the iteration of the mapping. This exponential growth rate is called the stretch factor, it is often more convenient to deal with the stretch factor instead of the topological entropy itself.
Through the work of David Fried, combined with another theorem of Thurston, it has been known that the “stretch factor” of a surface mapping is either (i) zero or (ii) a number which has an interesting algebraic property. An algebraic integer is a root of a monic polynomial with integer coefficients. (Note: the word ‘monic’ means the leading term has a coefficient of 1. The minimal polynomial of an algebraic integer is the one with the smallest degree among all such polynomials. The concept of ‘algebraic integer’ is a generalization of that of ‘integer’, since any given integer n is a root of the polynomial x-n=0, so it is an algebraic integer. An algebraic integer is called a “unit” if its reciprocal is also an algebraic integer.
Certain classes of algebraic integers such as Salem numbers, Pisot number, and Perron numbers have been the subjected more study in the literature. These classes are defined in terms of the distribution of the other roots of the minimal polynomial of the number. A positive algebraic integer L is called bi-Perron if all roots of the minimal polynomial smaller than L and larger than or equals to 1/L in absolute value. What Fried showed is that the stretch factor of a surface mapping is either (i) 0 or (ii) a bi-Perron algebraic unit. It has been conjectured that every bi-Perron algebraic unit is the stretch factor of some surface mapping, i.e., the converse to Fried’s theorem also holds.
One can test this conjecture using a probabilistic method. For any fixed positive number R, one can estimate the proportion of the stretch factors of mappings on the surface of genus n among the bi-Perron algebraic units of degree m, and then see how this proportion changes as R increases to infinity.
The researchers at KAIST led by Hyungryul Baik of the Department of Mathematical Sciences showed that this proportion converges to zero unless n is significantly bigger than m. The outline of the proof has two steps: first, one estimates the number of the bi-Perron algebraic units of degree m which are smaller than a given number R using complex analy, then, one can deduce an asymptotic upper bound on the number of the stretch factors of mappings on the surface of genus n which are smaller than R using a remarkable result from Eskin-Mirzakhani.
One can interpret this result as follows: even if Fried’s conjecture is true, it would be very difficult to verify it in the sense that to realize a given bi-Perron algebraic unit as the stretch factor of a surface mapping, one needs to consider a surface of very high genus.
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* lab webpage : http://www.hbaik.org/