Examples of type ${\rm III}_{1}$ inhomogenous Markov shifts supported on topological Markov shifts

Lyapunov optimization for non-generic one-dimensional expanding Markov maps

For a non-generic, yet dense subset of $C^1$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new $C^1$ perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

Projection of Markov measures may be Gibbsian

We study the induced measure obtained from a 1-step Markov measure, supported by a topological Markov chain, after the mapping of the original alphabet onto another one. We give sufficient conditions for the induced measure to be a Gibbs measure (in the sense of Bowen) when the factor system is again a topological Markov chain. This amounts to constructing, when it does exist, the induced potential and proving its Holder continuity. This is achieved through a matrix method. We provide examples and counterexamples to illustrate our results.

On Transfer Operators and Maps with Random Holes

We study Markov interval maps with random holes. The holes are not necessarily elements of the Markov partition. Under a suitable, and physically relevant, assumption on the noise, we show that the transfer operator associated with the random open system can be reduced to a transfer operator associated with the closed deterministic system. Exploiting this fact, we show that the random open system admits a unique (meaningful) absolutely continuous conditionally stationary measure. Moreover, we prove the existence of a unique probability equilibrium measure supported on the survival set, and we …

连续函数在非马尔可夫移位上的遍历优化

Ergodic optimization for continuous functions on non-Markov shifts

本文研究连续函数在非马尔可夫移位上的遍历优化问题，揭示了函数空间的划分特性，并推广了Morris和Shinoda的成果，应用于多种非马尔可夫符号动力系统。

A renewal scheme for non-uniformly hyperbolic semiflows

We investigate a renewal scheme for non-uniformly hyperbolic semiflows that closely resembles the renewal scheme developed in the discrete time case, in order to obtain sharp estimates for the correlation function. Also, the involved observables are supported on a flow-box of unbounded length. The present abstract setting does not require the use of Markov structure. However, the classes of examples covered here are rather restrictive. In these examples, it is easier to exploit the full force of the method and get optimal results for observables supported on finite length flow-boxes.

无穷远处的等分布与热力学

Equidistribution and thermodynamics at infinity

该研究针对可数马尔可夫移位及悬挂半流证明水平2大偏差上界，对强正递归位势建立加权经验测度等分布，应用于区间映射与高斯映射，得到边界点测度等分布及相关结果。