An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions

The Infimum of Lipschitz Constants in the Conjugacy Class of an Interval Map

How can we interpret the infimum of Lipschitz constants in a conjugacy class of interval maps? For positive entropy maps, the exponential of the topological entropy gives a well-known lower bound. We show that for piecewise monotone interval maps as well as for $C^{\infty}$ interval maps, these two quantities are equal, but for countably piecewise monotone maps, the inequality can be strict. Moreover, in the topologically mixing and Markov case, we characterize the infimum of Lipschitz constants as the exponential of the Salama entropy of a certain reverse Markov chain associated with the map.…

Length-expanding Lipschitz maps on totally regular continua

The tent map is an elementary example of an interval map possessing many interesting properties, such as dense periodicity, exactness, Lipschitzness and a kind of length-expansiveness. It is often used in constructions of dynamical systems on the interval/trees/graphs. The purpose of the present paper is to construct, on totally regular continua (i.e. on topologically rectifiable curves), maps sharing some typical properties with the tent map. These maps will be called length-expanding Lipschitz maps, briefly LEL maps. We show that every totally regular continuum endowed with a suitable metric…

具有平坦区间的某些圆周映射的$C^1$-微分同胚类

$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval

研究平坦段边界奇点次数不同的圆周映射类，证明其拓扑共轭若为双Lipschitz同胚则必为$C^1$微分同胚，从而两类等价。

A Lipschitz refinement of the Bebutov--Kakutani dynamical embedding theorem

We prove that an $\mathbb{R}$-action on a compact metric space embeds equivariantly in the space of one-Lipschitz functions $\mathbb{R}\to[0,1]$ if its fixed point set can be topologically embedded in the unit interval. This is a refinement of the classical Bebutov--Kakutani theorem (1968).

Bounded type interval exchange maps

Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to $r$-ball of the initial point is bounded below. In this paper, we consider how the bounded type condition of irrational is generalized into interval exchange maps.

Quantitative weak mixing for interval exchange transformations

We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok-Stepin, Veech, and Avila-Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have…