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In this paper, we study intermittent behaviors of coupled piecewise-expanding map lattices with two nodes and a weak coupling. We show that the successive phase transition between ordered and disordered phases occurs for almost every orbit. That is, we prove $\liminf_{n\rightarrow \infty}| x_1(n)-x_2(n)|=0$ and $\limsup_{n\rightarrow \infty}| x_1(n)-x_2(n)|\ge c_0>0$, where $x_1(n), x_2(n)$ correspond to the coordinates of two nodes at the iterative step $n$. We also prove the same conclusion for weakly coupled tent-map lattices with any multi-nodes.

We establish necessary and sufficient conditions for consistent root reconstruction in continuous-time Markov models with countable state space on bounded-height trees. Here a root state estimator is said to be consistent if the probability that it returns to the true root state converges to 1 as the number of leaves tends to infinity. We also derive quantitative bounds on the error of reconstruction. Our results answer a question of Gascuel and Steel [GS10] and have implications for ancestral sequence reconstruction in a classical evolutionary model of nucleotide insertion and deletion [TKF91].

We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ℂ is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$ ,$β$>0, and$λ$∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not$β$-porous in scale$λ$^{$n$}for$n$from a set with positive density amongst natural numbers.

We compute the one-loop effective action of two D0-branes in the matrix model for a cosmological background, and find vanishing static potential. However, there is a non-vanishing v2 term not predicted in a supergravity calculation. This term is complex and signals an instability of the two D0-brane system, it may also indicate that the matrix model is incorrect.