scholarly journals Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

scholarly journals One-dimensional game of life and its growth functions

1992 ◽  
Vol 15 (3) ◽  
pp. 499-508
Author(s):  
Mohammad H. Ahmadi
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Formal Power Series ◽  
Infinite Product ◽  
Growth Function ◽  
The Other ◽  
Arbitrary Integer ◽  
One Dimensional ◽  
Growth Functions ◽  
Formal Power

We start with finitely many1's and possibly some0's in between. Then each entry in the other rows is obtained from the Base2sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Defined1,jrecursively for1, a non-negative integer, andjan arbitrary integer by the rules:d0,j={1     for   j=0,k         (I)0   or   1   for   0<j<kd0,j=0   for   j<0   or   j>k              (II)di+1,j=di,j+1(mod2)   for   i≥0.      (III)Now, if we interpret the number of1's in rowias the coefficientaiof a formal power series, then we obtain a growth function,f(x)=∑i=0∞aixi. It is interesting that there are cases for which this growth function factors into an infinite product of polynomials. Furthermore, we shall show that this power series never represents a rational function.

scholarly journals Engel Series Expansions of Laurent Series and Hausdorff Dimensions

2003 ◽  
Vol 75 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Jun Wu
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Laurent Series ◽  
Series Expansions ◽  
Formal Laurent Series ◽  
Hausdorff Dimensions ◽  
Formal Power

AbstractFor any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

scholarly journals Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Laurent Poinsot
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Semidirect Product ◽  
Hopf Algebras ◽  
Formal Series ◽  
Product Structure ◽  
Locally Finite ◽  
Commutative Algebras ◽  
Affine Groups ◽  
Formal Power

A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.

Sign in / Sign up

Export Citation Format

Share Document