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Register a new userThe Mather measure and a Large Deviation Principle for the Entropy Penalized Method
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We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $mu$ is unique and the support of $mu$ is the Aubry set). Consider, for each value of $epsilon $ and h, the entropy penalized Mather problem $min {int_{tntimesrn} L(x,v)dmu(x,v)+epsilon S[mu]},$ where the entropy S is given by $S[mu]=int_{tntimesrn}mu(x,v)lnfrac{mu(x,v)}{int_{rn}mu(x,w)dw}dxdv,$ and the minimization is performed over the space of probability densities $mu(x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E. Valdinoci that there exists a minimizing measure $mu_{epsilon, h}$ which converges to the Mather measure $mu$. We show a LDP $lim_{epsilon,hto0} epsilon ln mu_{epsilon,h}(A),$ where $Asubset mathbb{T}^Ntimesmathbb{R}^N$. The deviation function I is given by $I(x,v)= L(x,v)+ ablaphi_0(x)(v)-bar{H}_{0},$ where $phi_0$ is the unique viscosity solution for L.
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In this paper we present an upper bound for the decay of correlation for the stationary stochastic process associated with the Entropy Penalized Method. Let $L(x, v):Tt^ntimesRr^nto Rr$ be a Lagrangian of the form L(x,v) = {1/2}|v|^2 - U(x) + < P, v>. For each value of $epsilon $ and $h$, consider the operator Gg[phi](x):= -epsilon h {ln}[int_{re^N} e ^{-frac{hL(x,v)+phi(x+hv)}{epsilon h}}dv], as well as the reversed operator bar Gg[phi](x):= -epsilon h {ln}[int_{re^N} e^{-frac{hL(x+hv,-v)+phi(x+hv)}{epsilon h}}dv], both acting on continuous functions $phi:Tt^nto Rr$. Denote by $phi_{epsilon,h} $ the solution of $Gg[phi_{epsilon,h}]=phi_{epsilon,h}+lambda_{epsilon,h}$, and by $bar phi_{epsilon,h} $ the solution of $bar Gg[phi_{epsilon,h}]=bar phi_{epsilon,h}+lambda_{epsilon,h}$. In order to analyze the decay of correlation for this process we show that the operator $ {cal L} (phi) (x) = int e^{- frac{h L (x,v)}{epsilon}} phi(x+h v) d v,$ has a maximal eigenvalue isolated from the rest of the spectrum.
We initiate a study of large deviations for block model random graphs in the dense regime. Following Chatterjee-Varadhan(2011), we establish an LDP for dense block models, viewed as random graphons. As an application of our result, we study upper tail large deviations for homomorphism densities of regular graphs. We identify the existence of a symmetric phase, where the graph, conditioned on the rare event, looks like a block model with the same block sizes as the generating graphon. In specific examples, we also identify the existence of a symmetry breaking regime, where the conditional structure is not a block model with compatible dimensions. This identifies a reentrant phase transition phenomenon for this problem---analogous to one established for Erdos-Renyi random graphs (Chatterjee-Dey(2010), Chatterjee-Varadhan(2011)). Finally, extending the analysis of Lubetzky-Zhao(2015), we identify the precise boundary between the symmetry and symmetry breaking regime for homomorphism densities of regular graphs and the operator norm on Erdos-Renyi bipartite graphs.
We establish the (level-1) large deviation principles for three kinds of means associated with the backward continued fraction expansion. We show that: for the harmonic and geometric means, the rate functions vanish exactly at one point; for the arithmetic mean, it is completely degenerate, vanishing at every point in its effective domain. Our method of proof employs the thermodynamic formalism for finite Markov shifts, and a multifractal analysis for the Renyi map generating the backward continued fraction digits. We completely determine the class of unbounded arithmetic functions for which the rate functions vanish at every point in unbounded intervals.
Let $Lambda$ be a complex manifold and let $(f_lambda)_{lambdain Lambda}$ be a holomorphic family of rational maps of degree $dgeq 2$ of $mathbb{P}^1$. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdins bound of the volume of the image of a dynamical ball. Applying our technics to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of $mathbb{P}^k$.
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