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Thomas Jordan

Publications and source records attributed to Thomas Jordan.

30 records · Page 2Linked to original sources
arXiv2012

Dimension of self-affine sets with holes

In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.

math.DS↗
arXiv2012

Phase transitions for suspension flows

This paper is devoted to study thermodynamic formalism for suspension flows defined over countable alphabets. We are mostly interested in the regularity properties of the pressure function. We establish conditions for the pressure function to be real analytic or to exhibit a phase transition. We also construct an example of a potential for which the pressure has countably many phase transitions.

math.DS↗
arXiv2011

Multifractal analysis of Birkhoff averages for countable Markov maps

In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain regularity assumptions on the potential the Birkhoff spectrum is real analytic. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.

math.DS↗
arXiv2010

Multifractal structure of Bernoulli convolutions

Let $ν_λ^p$ be the distribution of the random series $\sum_{n=1}^\infty i_n λ^n$, where $i_n$ is a sequence of i.i.d. random variables taking the values 0,1 with probabilities $p,1-p$. These measures are the well-known (biased) Bernoulli convolutions. In this paper we study the multifractal spectrum of $ν_λ^p$ for typical $λ$. Namely, we investigate the size of the sets \[ Δ_{λ,p}(α) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log ν_λ^p(B(x,r))}{\log r} =α\right\}. \] Our main results highlight the fact that for almost all, and in some cases all, $λ$ in an appropriate range, $Δ_{λ,p}(α)$ is nonempty and, moreover, has positive Hausdorff dimension, for many values of $α$. This happens even in parameter regions for which $ν_λ^p$ is typically absolutely continuous.

math.DS↗
arXiv2010

Increasing digit subsystems of infinite iterated function systems

We consider infinite iterated function systems $\{f_i\}_{i=1}^{\infty}$ on $[0,1]$ with a polynomially increasing contraction rate. We look at subsets of such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ f_{i_3}\circ...$ if $i_n>Φ(i_{n-1})$ for certain increasing functions $Φ:\mathbb{N}\rightarrow\mathbb{N}$. We compute both the Hausdorff and packing dimensions of such sets. Our results generalize work of Ramharter which shows that the set of continued fractions with strictly increasing digits has Hausdorff dimension 1/2.

math.DS↗
arXiv2009

Multifractal analysis for Bedford-McMullen carpets

In this paper we compute the multifractal analysis for local dimensions of Bernoulli measures supported on the self-affine carpets introduced by Bedford-McMullen. This extends the work of King where the multifractal analysis is computed with strong additional separation assumptions.

math.DS↗
arXiv2009

The Hausdorff dimension of the projections of self-affine carpets

We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $Λ$ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of $Λ$ in a non-principal direction has Hausdorff dimension $\min(γ,1)$, where $γ$ is the Hausdorff dimension of $Λ$. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.

math.DS↗
arXiv2009

Multifractal analysis of weak Gibbs measures for non-uniformly expanding C^1 maps

We consider the local dimension spectrum of a weak Gibbs measure on a C^1 non-uniformly hyperbolic system of Manneville- Pomeau type. We present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.

math.DS↗
arXiv2008

Higher Order Birkhoff Averages

There are well-known examples of dynamical systems for which the Birkhoff averages with respect to a given observable along some or all of the orbits do not converge. It has been suggested that such orbits could be classified using higher order averages. In the case of a bounded observable, we show that a classical result of G.H. Hardy implies that if the Birkhoff averages do not converge, then neither do the higher order averages. If the Birkhoff averages do not converge then we may denote by $[α_k,β_k]$ the limit set of the $k$-th order averages. The sequence of intervals thus generated is nested: $[α_{k+1},β_{k+1}] \subset [α_k,β_k]$. We can thus make a distinction among nonconvergent Birkhoff averages; either: B1. $\cap_{k=1}^\infty [α_k,β_k]$ is a point $B_\infty$, or, B2. $\cap_{k=1}^\infty [α_k,β_k]$ is a non-trivial interval $[α_\infty,β_\infty]$. We give characterizations of the types B1 and B2 in terms of how slowly they oscillate and we give examples that exhibit both behaviours B1 and B2 in the context of full shifts on finite symbols and "Bowen's example". For finite full shifts, we show that the set of orbits with type B2 behaviour has full topological entropy.

math.DS↗
arXiv2008

Sets of non-differentiability for conjugacies between expanding interval maps

We study differentiability of topological conjugacies between expanding piecewise $C^{1+ε}$ interval maps. If these conjugacies are not $C^1$, then they have zero derivative almost everywhere. We obtain the result that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Using multifractal analysis and thermodynamic formalism, we show that this Hausdorff dimension is explicitly determined by the Lyapunov spectrum. Moreover, we show that these results give rise to a "rigidity dichotomy" for the type of conjugacies under consideration.

math.DS↗
arXiv2003

Low-Cost Data Acquisition Card for School-Network Cosmic Ray Detectors

The Cosmic Ray Observatory Project (CROP) at University of Nebraska/Lincoln and the Washington Area Large-scale Time coincidence Array (WALTA) at University of Washington/Seattle are among several outreach projects siting cosmic-ray detectors at local high schools in cities around North America, to study the origins and interactions of high-energy cosmic rays. In a collaboration between QuarkNet, the outreach program based at Fermilab, CROP, and WALTA, a low-cost data acquisition electronics card has been developed to collect and synchronize the data from each detector site. The cost for each card is under US$500 for parts, functionally replacing much more expensive electronics crates and modules at each high school site. The card has 4 analog discriminator inputs for photo-multiplier tube signals, a 4-channel Time-to-Digital converter for local coincidence and time-over-threshold measurements at 0.75 ns resolution, programmable trigger logic via a CPLD and microcontroller, and a built-in low-cost GPS receiver/antenna module (via external cable) to provide event trigger time stamps at better than 100 ns accuracy. Temperature sensors and a barometer are also integrated to record environmental data along with the counter data. The card connects to any PC or laptop via a standard RS-232 serial port for data output and control. The microcontroller and CPLD are field programmable and therefore make the card functionality flexible and easy to upgrade.

physics.ins-det↗