Polynomial Entropy Research Articles

On Polynomial Entropy Of Induced Maps On Symmetric Products

On Polynomial Entropy Of Induced Maps On Symmetric Products

Polynomial Entropy and Polynomial Torsion for Fibered Systems

Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber,which measures the “infinitesimal variation” of the dynamics between the fiber and the neighboring ones.This gives rise to an (upper semicontinous) torsion function,defined on the base of the system, which is a newC^{0} (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum ofthe torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems.We examine the relevance of these results in the context of integrable Hamiltoniansystems or diffeomorphisms, with the particular cases of C^{0}-integrable twist maps on the annulus and geodesic flows.Finally, we bound from below the polynomial entropy of ell-modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.

Categorical action filtrations via localization and the growth as a symplectic invariant

Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.

Polynomial Entropy of Induced Maps of Circle and Interval Homeomorphisms

We compute the polynomial entropy of the induced maps on hyperspace for a homeomorphism $f$ of an interval or a circle with finitely many non-wandering points.

Variational principle for polynomial entropy on subsets of free semigroup actions

We investigate the relations between Bowen polynomial entropy on an arbitrary subset and local measure theoretic polynomial entropy of Borel probability measure for finitely generated semigroup actions. Also, we establish a variational principle for polynomial entropy on compact subsets in the context of free semigroup actions.

Polynomial Entropy of Amenable Group Actions for Noncompact Sets

Polynomial Entropy of Amenable Group Actions for Noncompact Sets

Polynomial entropy of Morse-Smale diffeomorphisms on surfaces

A classical problem in dynamical systems is to measure the complexity of a map in terms of its orbits, and one of the main concepts used to achieve this goal is entropy. Nonetheless, many interesting families of dynamical systems have every element with zero-entropy. One of these are Morse-Smale diffeomorphisms. In this work, we compute the generalized entropy of Morse-Smale diffeomorphisms on surfaces, based on which we deduce their polynomial entropy. We also apply our technique to compute the dispersion of the orbits of maps on the border of chaos with mild dissipation.

Polynomial entropy of nonautonomous dynamical systems for noncompact sets

This paper is devoted to the study of the Bowen polynomial entropy for nonautonomous dynamical systems, which is an extension of the classical definition of Bowen topological entropy. We establish a variational principle for polynomial entropy on compact subsets in the context of nonautonomous dynamical systems. Also, we show that the Bowen topological entropy can be determined by the local entropies of measures for nonautonomous dynamical systems, which extends Ma and Wen's result [33].

The core entropy for polynomials of higher degree

As defined by W. Thurston, the core entropy of a polynomial is the entropy of the restriction to its Hubbard tree. For each d \geq 2 , we study the core entropy as a function on the parameter space of polynomials of degree d , and prove it varies continuously both as a function of the combinatorial data and of the coefficients of the polynomials. This confirms a conjecture of W. Thurston.

Polynomial Entropy of Subsets for Free Semigroup Actions

Polynomial Entropy of Subsets for Free Semigroup Actions

Polynomial Entropy of the Logistic Map

We study the polynomial entropy of the logistic map depending on a parameter, and we calculate it for almost all values of the parameter. We show that polynomial entropy distinguishes systems with a low complexity (i.e. for which the topological entropy vanishes).

A survey on computing the topological entropy of cubic polynomials

In this paper we discuss two different existing algorithms for computing topological entropy and we implement one of them in order to compute the isentropes for cubic polynomials.

Categorical polynomial entropy

For classical dynamical systems, the polynomial entropy serves as a refined invariant of the topological entropy. In the setting of categorical dynamical systems, that is, triangulated categories endowed with an endofunctor, we develop the theory of categorical polynomial entropy, refining the categorical entropy defined by Dimitrov, Haiden, Katzarkov, and Kontsevich. We justify this notion by showing that for an automorphism of a smooth projective variety, the categorical polynomial entropy of the pullback functor on the derived category coincides with the polynomial growth rate of the induced action on cohomology. We also establish in general a Yomdin-type lower bound for the categorical polynomial entropy of an endofunctor in terms of the induced endomorphism on the numerical Grothendieck group of the category. As examples, we compute the categorical polynomial entropy for some standard functors like shifts, Serre functors, tensoring line bundles, automorphisms, spherical twists, P-twists, and so on, illustrating clearly how categorical polynomial entropy refines the study of categorical entropy and enables us to study the phenomenon of categorical trichotomy. A parallel theory of polynomial mass growth rate is developed in the presence of Bridgeland stability conditions.

Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials

Let μ be a measure from Szegő class on the unit circle T and let {fn} be the family of Schur functions generated by μ. In this paper, we prove a version of the classical Szegő's formula, which controls the oscillation of fn on T for all n⩾0. Then, we focus on an analog of Lusin's conjecture for polynomials {φn} orthogonal with respect to measure μ and prove that pointwise convergence of {|φn|} almost everywhere on T is equivalent to a certain condition on zeroes of φn.

Polynomial Entropy for Interval Maps and Lap Number

We prove an upper bound for the polynomial entropy of continuous, piecewise monotone maps of the interval, according to the number of intervals of monotonicity of its iterates. We give examples that show that this inequality is sharp. As a direct consequence of this inequality, the polynomial entropy of monotone, continuous, interval maps is always less than or equal to one. We give examples where we can also obtain lower bounds. We also prove analogous inequality in terms of total variations of the iterates of these interval maps. Also, this inequality is sharp.

Automorphisms of compact Kähler manifolds with slow dynamics

We study the automorphisms of compact K\ahler manifolds having slow dynamics. By adapting Gromov's classical argument, we give an upper bound on the polynomial entropy and study its possible values in dimensions $2$ and $3$. We prove that every automorphism with sublinear derivative growth is an isometry ; a counter-example is given in the $C^{\infty}$ context, answering negatively a question of Artigue, Carrasco-Olivera and Monteverde on polynomial entropy. Finally, we classify minimal automorphisms in dimension $2$ and prove they exist only on tori. We conjecture that this is true for any dimension.

Continuity of Weighted Operators, MuckenhouptApWeights, and Steklov Problem for Orthogonal Polynomials

AbstractWe consider weighted operators acting on $L^p({\mathbb{R}}^d)$ and show that they depend continuously on the weight $w\in A_p({\mathbb{R}}^d)$ in the operator topology. Then, we use this result to estimate $L^p_w({\mathbb{T}})$ norm of polynomials orthogonal on the unit circle when the weight $w$ belongs to Muckenhoupt class $A_2({\mathbb{T}})$ and $p>2$. The asymptotics of the polynomial entropy is obtained as an application.

Core Entropy of Quadratic Polynomials

We give a combinatorial definition of “core entropy” for quadratic polynomials as the growth exponent of the number of certain precritical points in the Julia set (those that separate the $$\alpha $$ fixed point from its negative). This notion extends known definitions that work in cases when the polynomial is postcritically finite or when the topology of the Julia set has good properties, and it applies to all quadratic polynomials in the Mandelbrot set. We prove that core entropy is continuous as a function of the complex parameter. In fact, we model the Julia set as an invariant quadratic lamination in the sense of Thurston: this depends on the external angle of a parameter in the boundary of the Mandelbrot set, and one can define core entropy directly from the angle in combinatorial terms. As such, core entropy is continuous as a function of the external angle. Moreover, we prove a conjecture of Giulio Tiozzo about local and global maxima of core entropy as a function of external angles: local maxima are exactly dyadic angles, and the unique global maximum within any wake occurs at the dyadic angle of lowest denominator. We also describe where local minima occur. An appendix by Wolf Jung relates different concepts of core entropy and biaccessibility dimension and thus shows that biaccessibility dimension is continuous as well.

Weak Limits of the Measures of Maximal Entropy for Orthogonal Polynomials

In this paper we study the sequence of orthonormal polynomials $\{P_n(\mu; z)\}$ defined by a probability measure $\mu$ with non-polar compact support $S(\mu)\subset\mathbb C$. We show that the support of any weak* limit of the sequence of measures of maximal entropy $\omega_n$ for $P_n$ is contained in the polynomial-convex hull of $S(\mu)$. And for $n$-th root regular measures the $\omega_n$ converge weak* to the equilibrium measure on $S(\mu)$.

On Thurston’s core entropy algorithm

The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant extending topological entropy for real maps to complex polynomials, whence providing a new tool to study the parameter space of polynomials. The base is a combinatorial algorithm allowing for the computation of the core entropy given by Thurston but without supplying a proof. In this paper, we will describe his algorithm and prove its validity.