A number theoretic question arising in the geometry of plane curves and in billiard dynamics

We prove that if $\rho\neq1/2$ is a rational number between zero and one, then there is no integer $n>1$ such that $$n\tan(\pi\rho)=\tan(n\pi\rho).$$ This has interpretations both in the theory of bicycle curves and that of mathematical billiards

Counting generic measures for a subshift of linear growth

In 1984 Boshernitzan proved an upper bound on the number of ergodic measures for a minimal subshift of linear block growth and asked if it could be lowered without further assumptions on the shift. We answer this question, showing that Boshernitzan's bound is sharp. We further prove that the same bound holds for the, a priori, larger set of nonatomic generic measures, and that this bound remains valid even if one drops the assumption of minimality. Applying these results to interval exchange transformations, we give an upper bound on the number of nonatomic generic measures of a minimal IET, answering a question recently posed by Chaika and Masur

Free ergodic Z2\mathbb{Z}^2-systems and complexity

Using results relating the complexity of a two dimensional subshift to its periodicity, we obtain an application to the well-known conjecture of Furstenberg on a Borel probability measure on $[0,1)$ which is invariant under both $x\mapsto px \pmod 1$ and $x\mapsto qx \pmod 1$, showing that any potential counterexample has a nontrivial lower bound on its complexity

Positive entropy equilibrium states

For transitive shifts of finite type, and more generally for shifts with specification, it is well-known that every equilibrium state for a Holder continuous potential has positive entropy as long as the shift has positive topological entropy. We give a non-uniform specification condition under which this property continues to hold, and demonstrate that it does not necessarily hold for other non-uniform versions of specification that have been introduced elsewhere.Comment: 17 pages, corrected proof in Section 4.

The automorphism group of a minimal shift of stretched exponential growth

The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the direct sum of countably many copies of $\mathbb{Z}$. In contrast, the group of automorphisms of a symbolic system of zero entropy seems to be highly constrained. Our main result is that the automorphism group of any minimal subshift of stretched exponential growth with exponent $<1/2$, is amenable (as a countable discrete group). For shifts of polynomial growth, we further show that any finitely generated, torsion free subgroup of Aut(X) is virtually nilpotent

The automorphism group of a shift of slow growth is amenable

Suppose $(X,\sigma)$ is a subshift, $P_X(n)$ is the word complexity function of $X$, and ${\rm Aut}(X)$ is the group of automorphisms of $X$. We show that if $P_X(n)=o(n^2/\log^2 n)$, then ${\rm Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_X(n)=o(n^2)$, then ${\rm Aut}(X)$ can never contain a nonabelian free semigroup (and, in particular, can never contain a nonabelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a semigroup

Realizing ergodic properties in zero entropy subshifts

A subshift with linear block complexity has at most countably many ergodic measures, and we continue of the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity is arbitrarily close to linear but has uncountably many ergodic measures, we show that this behavior fails as soon as the block complexity is superlinear. With a different construction, we show that there exists a minimal subshift with an ergodic measure whose slow entropy grows slower than any given rate tending to infinitely but faster than any other rate majorizing this one yet still growing subexponentially. These constructions lead to obstructions in using subshifts in applications to properties of the prime numbers and in finding a measurable version of the complexity gap that arises for shifts of sublinear complexity

The automorphism group of a shift of subquadratic growth

For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length $n$. When this complexity grows exponentially, the automorphism group has been shown to be large for various classes of subshifts. In contrast, we show that subquadratic growth of the complexity implies that for a topologically transitive shift $X$, the automorphism group \Aut(X) is small: if $H$ is the subgroup of \Aut(X) generated by the shift, then \Aut(X)/H is periodic

Nonexpansive Z^2 subdynamics and Nivat's conjecture

For a finite alphabet \A and \eta\colon \Z\to\A, the Morse-Hedlund Theorem states that $\eta$ is periodic if and only if there exists $n\in\N$ such that the block complexity function $P_\eta(n)$ satisfies $P_\eta(n)\leq n$, and this statement is naturally studied by analyzing the dynamics of a $\Z$-action associated to $\eta$. In dimension two, we analyze the subdynamics of a \ZZ-action associated to \eta\colon\ZZ\to\A and show that if there exist $n,k\in\N$ such that the $n\times k$ rectangular complexity $P_{\eta}(n,k)$ satisfies $P_{\eta}(n,k)\leq nk$, then the periodicity of $\eta$ is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist $n,k\in\N$ such that $P_{\eta}(n,k)\leq \frac{nk}{2}$, then $\eta$ is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words

Complexity and directional entropy in two dimensions

We study the directional entropy of the dynamical system associated to a $\Z^2$ configuration in a finite alphabet. We show that under local assumptions on the complexity, either every direction has zero topological entropy or some direction is periodic. In particular, we show that all nonexpansive directions in a $\Z^2$ system with the same local assumptions have zero directional entropy