The critical dimension for -measures

2015 ◽  
Vol 37 (3) ◽  
pp. 824-836
Author(s):  
DANIEL F. MANSFIELD ◽  
ANTHONY H. DOOLEY
Keyword(s):  
Dynamical System ◽  
Growth Rate ◽  
Critical Dimension ◽  
Individual Factors ◽  
Invariant Property ◽  
Von Neumann ◽  
Asymptotic Growth ◽  
Asymptotic Bound ◽  
Asymptotic Growth Rate

The critical dimension of an ergodic non-singular dynamical system is the asymptotic growth rate of sums of consecutive Radon–Nikodým derivatives. This has been shown to equal the average coordinate entropy for product odometers when the size of individual factors is bounded. We extend this result to $G$-measures with an asymptotic bound on the size of individual factors. Furthermore, unlike von Neumann–Krieger type, the critical dimension is an invariant property on the class of ergodic $G$-measures.

Growth in Baumslag–Solitar groups II: The Bass–Serre tree

2020 ◽  
Vol 30 (02) ◽  
pp. 339-378
Author(s):  
Jared Adams ◽  
Eric M. Freden
Keyword(s):  
Growth Rate ◽  
Minimal Length ◽  
Geometric Object ◽  
Asymptotic Growth ◽  
Growth Series ◽  
Context Sensitive ◽  
Counting Algorithms ◽  
Self Similar ◽  
Context Free ◽  
Asymptotic Growth Rate

Denote the Baumslag–Solitar family of groups as [Formula: see text]). When [Formula: see text] we study the Bass–Serre tree [Formula: see text] for [Formula: see text] as a geometric object. We suggest that the irregularity of [Formula: see text] is the principal obstruction for computing the growth series for the group. In the particular case [Formula: see text] we exhibit a set [Formula: see text] of normal form words having minimal length for [Formula: see text] and use it to derive various counting algorithms. The language [Formula: see text] is context-sensitive but not context-free. The tree [Formula: see text] has a self-similar structure and contains infinitely many cone types. All cones have the same asymptotic growth rate as [Formula: see text] itself. We derive bounds for this growth rate, the lower bound also being a bound on the growth rate of [Formula: see text].

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (03) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

scholarly journals Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process

1969 ◽  
Vol 10 (1-2) ◽  
pp. 231-235 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Growth Rate ◽  
Population Size ◽  
Branching Process ◽  
Expected Number ◽  
Lattice Distribution ◽  
Asymptotic Growth ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
Asymptotic Growth Rate

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (3) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

scholarly journals Some Local Asymptotic Laws for the Cauchy Process on the Line

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
A. Chukwuemeka Okoroafor
Keyword(s):  
Growth Rate ◽  
Sojourn Time ◽  
Escape Rate ◽  
Time Process ◽  
Rate Process ◽  
Monotone Functions ◽  
Asymptotic Growth ◽  
Asymptotic Size ◽  
Asymptotic Growth Rate ◽  
Cauchy Process

This paper investigates the lim inf behavior of the sojourn time process and the escape rate process associated with the Cauchy process on the line. The monotone functions associated with the lower asymptotic growth rate of the sojourn time are characterized and the asymptotic size of the large values of the escape rate process is developed.

On the asymptotic growth rate of some spanning trees embedded in

2011 ◽  
Vol 39 (1) ◽  
pp. 44-48 ◽  
Author(s):  
Pedro M.M. de Castro ◽  
Olivier Devillers
Keyword(s):  
Growth Rate ◽  
Spanning Trees ◽  
Asymptotic Growth ◽  
Asymptotic Growth Rate

scholarly journals The growth rate of trajectories of a quadratic differential

1990 ◽  
Vol 10 (1) ◽  
pp. 151-176 ◽  
Author(s):  
Howard Masur
Keyword(s):  
Growth Rate ◽  
Riemann Surface ◽  
Quadratic Differential ◽  
Compact Riemann Surface ◽  
Saddle Connection ◽  
Asymptotic Growth ◽  
Holomorphic Quadratic Differential ◽  
Asymptotic Growth Rate

AbstractSupposeqis a holomorphic quadratic differential on a compact Riemann surface of genusg≥ 2. Thenqdefines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at mostTis at most quadratic inT. An application is given to billiards.

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