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We show that there is a bijection between the renormalizations andproper completely invariant closed sets of expanding Lorenz map,which enables us to distinguish periodic and non-periodicrenormalizations.
By projecting back to the 1-d map, we are able to prove that it inherits nice statistical properties, including the large deviation principle, the exponential decay of correlations, as well as the almost sure invariance principle for the expanding map on a large class of observables.
Under a new small boundary pressure condition, we improve the estimate by establishing a variational principle for piecewise expanding maps and subadditive potentials.
The new result is that the coboundary can be chosen Lipschitz with a uniform control on the Lipschitz norm. In addition our result holds true for possibly non invertible and not transitive $ C^1 $ maps. We actually prove the main result in the setting of locally maximal hyperbolic sets for general $ C^1 $ map.
As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability.
Langevin dynamics are widely used in sampling high-dimensional, non-Gaussian distributions whose densities are known up to a normalizing constant. In particular, there is strong interest in unadjusted Langevin algorithms (ULA), which directly discretize Langevin dynamics to estimate expectations over the target distribution. We study the use of transport maps that approximately normalize a ...
L. Bers, An extremal problem for quasiconformal maps and a theorem by Thurston, Acta Math., 141 (1978), 73-98.doi: 10.1007/BF02545743. [7] A. Bufetov, Logarithmic asymptotics for the number of periodic orbits of the Teichmueller flow on Veech's space of zippered rectangles, Mosc. Math. J., 9 (2009), 245-261, back matter
We prove a conjecture of G.A. Margulis on the abundance of certainexceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizinga theory of modified Schmidt games, which are modifications of $(\\alpha,\\beta)$-games introduced by W. Schmidt in mid-1960s.
As a natural counterpart to Nakada's $ \alpha $-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1.
The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small.