Polynomial Semigroups Research Articles

On amenability and measure of maximal entropy for semigroups of rational maps: II

We compare dynamical and algebraic properties of semigroups of rational maps. In particular, we show a version of the Day-von Neumann conjecture and give a partial positive answer to “Sushkievich’s problem” for semigroups of rational maps. We relate these conjectures with Furstenberg’s [Formula: see text] problem and prove a coarse version of Furstenberg’s problem for semigroups of non-exceptional polynomials.

On amenable semigroups of rational functions

We characterize left and right amenable semigroups of polynomials of one complex variable with respect to the composition operation. We also prove a number of results about amenable semigroups of arbitrary rational functions. In particular, we show that under quite general conditions a semigroup of rational functions is amenable if and only if it is a subsemigroup of the centralizer of some rational function.

Dynamics of polynomial semigroups: measures, potentials, and external fields

In this paper, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh-Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Along the way, we establish the continuity of the logarithmic potential for the Dinh-Sibony measure, which might be of independent interest. We then use the F-functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.

Sharing a Measure of Maximal Entropy in Polynomial Semigroups

Abstract Let $P_1,P_2,\dots , P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or to Chebyshev polynomials, and $S$ the semigroup under composition generated by $P_1,P_2,\dots , P_k$. We show that all elements of $S$ share a measure of maximal entropy if and only if the intersection of principal left ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.

On intersections of polynomial semigroups orbits with plane lines

AbstractWe study intersections of orbits in polynomial semigroup dynamics with lines on the affine plane over a number field, extending previous work of D. Ghioca, T. Tucker, and M. Zieve (2008).

On algebraic integers of bounded house and preperiodicity in polynomial semigroup dynamics

We consider semigroup dynamical systems defined by several polynomials over a number field $\mathbb{K}$, and the orbit (tree) they generate at a given point. We obtain finiteness results for the set of preperiodic points of such systems that fall in the cyclotomic closure of $\mathbb{K}$. More generally, we consider the finiteness of initial points in the cyclotomic closure for which the orbit contains an algebraic integer of bounded house. This work extends previous results for classical obits generated by one polynomial over $\mathbb{K}$ obtained initially by Dvornicich and Zannier (for preperiodic points), and then by Chen and Ostafe (for roots of unity and elements of bounded house in orbits).

Semigroups whose Julia sets are Cantor targets

ABSTRACTWe consider semigroups generated by two rational functions whose Julia sets are Cantor targets. Noting that a Cantor target has no interior points, we construct a polynomial semigroup whose Julia set has no interior points and the Hausdorff dimension of whose Julia set is arbitrary close to 2.

The space of 2-generator postcritically bounded polynomial semigroups and random complex dynamics

We investigate the dynamics of 2-generator semigroups of polynomials with bounded planar postcritical set and associated random dynamics on the Riemann sphere. Also, we investigate the space B of such semigroups. We show that for a parameter h in the intersection of B, the hyperbolicity locus H and the closure of the disconnectedness locus (the space of parameters for which the Julia set is disconnected), the corresponding semigroup satisfies either the open set condition (and Bowen's formula) or that the Julia sets of the two generators coincide. Also, we show that for such a parameter h, if the Julia sets of the two generators do not coincide, then there exists a neighborhood U of h such that for each parameter in U, the Hausdorff dimension of the Julia set of the corresponding semigroup is strictly less than 2. Moreover, we show that the intersection of the connectedness locus and B∩H has dense interior. By using the results on the semigroups corresponding to these parameters, we investigate the associated functions which give the probability of tending to ∞ (complex analogues of the devil's staircase or Lebesgue's singular functions) and complex analogues of the Takagi function.

C2MP: Chebyshev chaotic map-based authentication protocol for RFID applications

Radio frequency identification (RFID) is a promising wireless sensor technology in the Internet of Things and can be applied for object identification. However, the security issues are still open challenges and should be addressed to achieve enhanced safeguard. Existing security solutions mainly apply logical operators, hash function, and other cryptographic primitives to design authentication schemes. In this paper, we propose a Chebyshev chaotic map-based authentication protocol (C2MP) for the RFID applications. Thereinto, Chebyshev polynomial's semigroup and chaotic properties are introduced for identity authentication and anonymous data transmission. The proposed C2MP owns the security properties including data integrity, authentication, anonymity, and session freshness. According to the BAN logic, security formal analysis is performed based on the messages formalization, initial assumptions, anticipant goals, and logic verification. It indicates that the proposed C2MP is suitable for universal RFID applications.

Recurrence analysis on Julia sets of semigroups of complex polynomials

We introduce a recurrence function in order to analyze the dynamics of semigroups of complex polynomials. We show that under a regularity hypothesis, the recurrence function is continuous in the complex plane. This is a new notion even for the case of a semigroup with just one generator.

Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets

We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with a bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation that are parameterized by the Cantor set, densely inside the Julia set of the semigroup. Moreover, we give a sufficient condition for a fiberwise Julia set Jγ to satisfy that Jγ is a Jordan curve but not a quasicircle, the unbounded component of C ^ ∖ J γ is a John domain and the bounded component of ℂ \ Jγ is not a John domain. We show that under certain conditions, a random Julia set is almost surely a Jordan curve, but not a quasicircle. Many new phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are found and systematically investigated.

Transversality family of expanding rational semigroups

We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2. We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each a∈C with |a|≠0,1, the family of small perturbations of the semigroup generated by {z2,az2} satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.

Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup G G of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map g ∈ G g \in G . In general, the Julia set of such a semigroup G G may be disconnected, and each Fatou component of such G G is either simply connected or doubly connected. In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of G . G. Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps g ∈ G g \in G are distributed within the Julia set of the entire semigroup G G . We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.

Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets

We investigate the dynamics ofsemigroups generated by a family of polynomial maps on the Riemann spheresuch that the postcritical set in the complex plane is bounded.The Julia set of such a semigroup may not be connected in general.We show that for such apolynomial semigroup,if $A$ and $B$ are two connected components of the Julia set,then one of $A$ and $B$ surrounds the other. From this,it is shown that each connected component of the Fatou set iseither simply or doubly connected.Moreover, we show that the Julia set of such a semigroupis uniformly perfect.An upper estimate of the cardinality of the set of allconnected components of the Julia set of such a semigroup is given.By using this, we givea criterion for the Julia setto be connected.Moreover, we show that for any $n\in N \cup \{ \aleph _{0}\} ,$there exists a finitely generated polynomial semigroup with boundedplanar postcritical setsuch that the cardinality of the set of all connected components of theJulia set is equal to $n.$Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomialsare found and systematically investigated.

Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

AbstractWe investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere $\CCI $) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups G such that G is generated by a compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. In one of the classes, we have that, for almost every sequence $\gamma \in \Gamma ^{\NN }$, the Julia set Jγ of γ is a Jordan curve but not a quasicircle, the unbounded component of $\CCI {\setminus } J_{\gamma }$ is a John domain, and the bounded component of $\CC {\setminus } J_{\gamma }$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups G such that the planar postcritical set of G is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.

Interaction cohomology of forward or backward self-similar systems

We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings U of the invariant set, where each U consists of (backward) images of the invariant set under elements of finite word length. We give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of semigroups of polynomials. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.

Image ofLp(ℝn)under the Hermite Semigroup

It is shown that the Hermite (polynomial) semigroup{e−tℍ:t>0}mapsLp(ℝn,ρ)into the space of holomorphic functions inLr(ℂn,Vt,p/2(r+ϵ)/2)for eachϵ>0, whereρis the Gaussian measure,Vt,p/2(r+ϵ)/2is a scaled version of Gaussian measure withr=pif1<p<2andr=p′if2<p<∞with1/p+1/p′=1. Conversely ifFis a holomorphic function which is in a “slightly” smaller space, namelyLr(ℂn,Vt,p/2r/2), then it is shown that there is a functionf∈Lp(ℝn,ρ)such thate−tℍf=F. However, a single necessary and sufficient condition is obtained for the image ofL2(ℝn,ρp/2)undere−tℍ,1<p<∞. Further it is shown that ifFis a holomorphic function such thatF∈L1(ℂn,Vt,p/21/2)orF∈Lm1,p(ℝ2n), then there exists a functionf∈Lp(ℝn,ρ)such thate−tℍf=F, wherem(x,y)=e−x2/(p−1)e4t+1e−y2/e4t−1and1<p<∞.

On a certain kind of polynomials of degree 4 with disconnected Julia set

For a polynomial of degree at least two, the Julia set and thefilled-in Julia set are either connected or have uncountably manycomponents. In the case that the Julia set of a polynomial of degree 4is neither connected nor totally disconnected, there exists ahomeomorphism between the set of all components of the filled-inJulia set and some subset of the corresponding symbol space.Furthermore the polynomial is topologically conjugate to the shiftmap via the homeomorphism. Moreover there exists ahomeomorphism between the Julia sets of the polynomial and that of acertain polynomial semigroup.

On Irreducibles in the Polynomial Semigroup

We consider the asymptotic behavior of the number of irreducible polynomials over a finite field in arithmetic progressions.

Random dynamics of polynomials and devil’s-staircase-like functions in the complex plane

We consider the dynamics of polynomial semigroups with bounded postcritical set and random dynamics of complex polynomials in the complex plane. A polynomial semigroup G is a semigroup generated by polynomials in one variable with the semigroup operation being functional composition. We show that if the postcritical set of G, that is the closure of the G-orbit of the union of any critical values of any generators of G, is bounded in the complex plane, then the space of components of the Julia set of G (Julia set is the set of points in the Riemann sphere C ¯ in which G is not normal) has a total order “⩽”, where for two compact connected sets K 1, K 2 in C ¯ , K 1 ⩽ K 2 indicates that K 1 = K 2, or K 1 is included in a bounded component of C ¯ ⧹ K 2 . Using the above result and combining it with the theory of random dynamics of complex polynomials, we consider the following: Let τ be a Borel probability measure in the space { g ∈ C [ z ] | deg ( g ) ⩾ 2 } with topology induced by the uniform convergence on the Riemann sphere C ¯ . We consider the i.i.d. random dynamics in C ¯ such that at every step we choose a polynomial according to the distribution τ. Let T ∞( z) be the probability of tending to ∞ ∈ C ¯ starting from the initial value z ∈ C ¯ and let G τ be the polynomial semigroup generated by the support of τ. Suppose that the support of τ is compact, the postcritical set of G τ is bounded in the complex plane and the Julia set of G τ is disconnected. Then, we show that (1) in each component U of the complement of the Julia set of G τ , T ∞∣ U equals a constant C U , (2) T ∞ : C ¯ → [ 0 , 1 ] is a continuous function on the whole C ¯ , and (3) if J 1, J 2 are two components of the Julia set of G τ with J 1 ⩽ J 2, then max z ∈ J 1 T ∞ ( z ) ⩽ min z ∈ J 2 T ∞ ( z ) . Hence T ∞ is similar to the devil’s-staircase function.