Natural extensions for piecewise affine maps via Hofbauer towers

分段单调映射的周期测度密度

Density of periodic measures for piecewise monotonic maps

本文证明了具有正度量熵的分段单调映射的遍历测度可由周期测度在弱-*意义下逼近，部分回答了Hofbauer-Raith猜想。

Birkhoff spectrum for piecewise monotone interval maps

For piecewise monotone interval maps we look at Birkhoff spectra for regular potential functions. This means considering the Hausdorff dimension of the set of points for which the Birkhoff average of the potential takes a fixed value. In the uniformly hyperbolic case we obtain complete results, in the case with parabolic behaviour we are able to describe the part of the sets where the lower Lyapunov exponent is positive. In addition we give some lower bounds on the full spectrum in this case. This is an extension of work of Hofbauer on the entropy and Lyapunov spectra.

Irregular sets for piecewise monotonic maps

For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $β$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith p…

具有可数分支的分段凸映射的 Frobenius-Perron 算子的拟紧性

Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches

本文利用 Lasota-Yorke 不等式证明了具有可数无穷分支的分段凸映射的 Frobenius-Perron 算子的拟紧性，并确立了其绝对连续不变测度的存在性及系统的强遍历性质。

Metrics on trees I. The tower algorithm for interval maps

We consider Milnor's "tower algorithm" in the space of piecewise monotone maps, an iterative algorithm on the space of metrics which unifies, on the one hand, Thurston's iterative scheme which converges to holomorphic models, and, on the other hand, the theory of piecewise linear models coming from kneading theory. We prove that the algorithm converges for unimodal maps of high entropy, and provide examples where it does not converge for unimodal maps of low entropy and multimodal maps of higher degree.

Young Towers for Product Systems

We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $ C^2 $ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.