S. Bhattacharya, T. Ward
Let X_1 and X_2 be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map X_1 to X_2 is affine (that is, X_2 is topologically rigid) if and only if the system X_2 has finite topological entropy.
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In this paper, we consider entropy spectra on topological Markov shifts. We prove that if two measure-preserving dynamical systems of Gibbs measures with Hölder continuous potentials are isomorphic, then their entropy spectra are the same. This result raises a new rigidity problem. We call this problem the weak rigidity problem, contrasting it with the strong rigidity problem proposed by Barreira and Saraiva. We give a complete answer to the weak rigidity problem for Markov measures on a topological Markov shift with an aperiodic transition matrix of size 2.
We classify complex projective surfaces with an automorphism of positive entropy for which the unique invariant measure of maximal entropy is absolutely continuous with respect to Lebesgue measure.
We study the rigidity properties of a class of algebraic Z^3-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide examples of non-isomorphic Z^3-actions with all their Z^2-sub-actions isomorphic. The proofs use lexicographic half-space entropies and total ergodicity along critical directions.
For a nonlinear Anosov diffeomorphism of the 2-torus, we present examples of measures so that the group of $μ$-preserving diffeomorphisms is, up to zero-entropy transformations, cyclic. For families of equilibrium states $μ$, we strengthen this to show that the group of $μ$-preserving diffeomorphism is virtually cyclic.
For any $C^\infty$, area-preserving Anosov diffeomorphism $f$ of a surface, we show that a suspension flow over $f$ is $C^\infty$-conjugate to a constant-time suspension flow of a hyperbolic automorphism of the two torus if and only if the volume measure is the measure with maximal entropy. We also show that the the metric entropy with respect to the volume measure and the topological entropy of suspension flow over Anosov diffeomorphisms on torus achieve all possible values. Our results fit into two programs related to entropy rigidity and flexibility of Anosov systems.
Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study a family of generalized Bowen-Series boundary maps and prove the following rigidity result: in this family the topological entropy is constant and depends only on the genus of the surface. We give an explicit formula for this entropy and show that the value of the topological entropy also stays constant in the Teichmüller space of the surface. The proofs use conjugation to maps of constant slope.
Continuity of isomorphisms applied to rigidity problems of entropy spectra
本文研究拓扑马尔可夫移位上吉布斯测度系统的同构性质，证明使同构由移位自同构诱导的2-局部常值函数集合包含满测度开集，并据此证明熵谱的强非刚性与弱非刚性等价。
Given an integer $k \geq 5$, and a $C^k$ Anosov flow $Φ$ on some compact connected $3$-manifold preserving a smooth volume, we show that the measure of maximal entropy (MME) is the volume measure if and only if $Φ$ is $C^{k-\varepsilon}$-conjugate to an algebraic flow, for $\varepsilon>0$ arbitrarily small. Besides the rigidity, we also study the entropy flexibility, and show that the metric entropy with respect to the volume measure and the topological entropy of suspension flows over Anosov diffeomorphisms on the $2$-torus achieve all possible values subject to natural normalizations. Moreov…
We introduce the notion of a one-way horseshoe and show that the polynomial entropy of an interval map is given by one-way horseshoes of iterates of the map, obtaining in such a way an analogue of Misiurewicz's theorem on topological entropy and standard `two-way' horseshoes. Moreover, if the map is of Sharkovskii type 1 then its polynomial entropy can also be computed by what we call chains of essential intervals. As a consequence we get a rigidity result that if the polynomial entropy of an interval map is finite, then it is an integer. We also describe the possible values of polynomial entr…
We set up a real entropy function $h_\Bbb{R}$ on the space $\mathcal{M}'_d$ of Möbius conjugacy classes of real rational maps of degree $d$ by assigning to each class the real entropy of a representative $f\in\Bbb{R}(z)$; namely, the topological entropy of its restriction $f\restriction_{\hat{\Bbb{R}}}$ to the real circle. We prove a rigidity result stating that $h_\Bbb{R}$ is locally constant on the subspace determined by real maps quasi-conformally conjugate to $f$. As examples of this result, we analyze real analytic stable families of hyperbolic and flexible Lattès maps with real coefficie…
Rigidité, expansion et entropie en dynamique non-archimédienne (Rigidity, expansion and entropy in non-Archimedean dynamics)
该研究在非阿基米德动力学中证明了有理映射的刚性性质，表明其拓扑熵为整数对数，并建立了李亚普诺夫指数与野分支位置的关系。
We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities
We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.
A deep analysis of the Lyapunov exponents, for stationary sequence of matrices going back to Furstenberg, for more general linear cocycles by Ledrappier and generalized to the context of non-linear cocycles by Avila and Viana, gives an invariance principle for invariant measures with vanishing central exponents. In this paper, we give a new criterium formulated in terms of entropy for the invariance principle and in particular, obtain a simpler proof for some of the known invariance principle results. As a byproduct, we study ergodic measures of partially hyperbolic diffeomorphisms whose cente…
We study a notion of entropy for probability measure preserving actions of finitely generated free groups, called f-invariant entropy, introduced by Lewis Bowen. In the degenerate case, the f-invariant entropy is negative infinity. In this paper we investigate the qualitative consequences of having finite f-invariant entropy. We find three main properties of such actions. First, the stabilizers occurring in factors of such actions are highly restricted. Specifically, the stabilizer of almost every point must be either trivial or of finite index. Second, such actions are very chaotic in the sen…
We prove entropy rigidity for finite volume strictly convex projective manifolds in dimensions $\geq 3$, generalizing the work of arXiv:1708.03983 to the finite volume setting. The rigidity theorem uses the techniques of Besson, Courtois, and Gallot's entropy rigidity theorem. It implies uniform lower bounds on the volume of any finite volume strictly convex projective manifold in dimensions $\geq 3$.
Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen--Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter. We present and sketch the proofs of two strikingly different results: topological entropy is constant in this entire family ("rigidity"), while measure-theoretic entropy varies within Teichmüller space, taking all values ("flexibility") between zero…
The original proof has a gap, and need extra hypothesis that the strong stable and strong unstable filiation both to be $C^1$. The argument is like the following: with the regularity, one can show that the weak-stable and weak-unstable foliation both to be $C^{1+Lip}$, and then following the same argument as in the paper one can conclude the proof. But this extra hypothesis seems implying that the flow has a contact structure. Then it will be only a result which improves the Foulon's proof on contract structure for $C^\infty$ regularity to $C^2$, not so interest.
Let $d > 1$, and let $(X,α)$ and $(Y,β)$ be two zero-entropy ${\mathbb{Z}}^d$-actions on compact abelian groups by $d$ commuting automorphisms. We show that if all lower rank subactions of $α$ and $β$ have completely positive entropy, then any measurable equivariant map from $X$ to $Y$ is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.
We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.
We investigate rigidity of measurable structure for higher rank abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a 0 entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.
An algebraic $Z^{d}$-action is an action of $Z^{d}$ on a compact abelian group $X$ by automorphisms of $X$. We prove that for $d \ge 8$, there exist mixing zero entropy algebraic $Z^{d}$-actions which do not exhibit isomorphism rigidity property.
Entropy Rigidity for Maximal Representations
本文研究了最大表示下的熵刚性问题，通过引入可测的格罗莫夫积和相关测度，证明了最大表示在熵方面的刚性性质。
We study the cohomological pressure introduced by R.Sharp (defined by using topological pressures of certain potentials of Anosov flows). In particular, we get the rigidity in the case that this pressure coincides with the metrical entropy, generalising related rigidity results of A.Katok and P. Foulon.
Entropy collapse versus entropy rigidity for Reeb and Finsler flows
该研究对比切触流形上Reeb流的拓扑熵可任意小的熵坍缩现象，与许多闭流形上归一化Finsler测地流拓扑熵有一致正下界的熵刚性性质，揭示两类流的熵行为差异。
For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson--Courtois--Gallot's entropy rigidity result to Hilbert geometries.
Assuming positive entropy we prove a measure rigidity theorem for higher rank actions on tori and solenoids by commuting automorphisms. We also apply this result to obtain a complete classification of disjointness and measurable factors for these actions.
Rigidity for $u$-Gibbs measures admitting Margulis systems
本文研究了u-吉布斯测度的熵刚性，证明在特定条件下雅可比数据为常数，并探讨了其在不同映射类型中的应用与结论。
Rigidity of Furstenberg entropy under ucp maps
本文证明了迹von Neumann代数的平稳W$^*$-扩张间保持态的M-双模ucp映射保持Furstenberg熵当且仅当诱导Radon-Nikodym因子同构，将该结果推广至局部紧群拟因子映射并证明熵分离，最终为唯一平稳Poisson边界建立刚性现象。
Given a surface $M$ and a Borel probability measure $ν$ on the group of $C^2$-diffeomorphisms of $M$, we study $ν$-stationary probability measures on $M$. Assuming the positivity of a certain entropy, the following dichotomy is proved: either the stable distributions for the random dynamics is non-random, or the measure is SRB. In the case that $ν$-a.e. diffeomorphism preserves a common smooth measure $m$, we show that for any positive-entropy stationary measure $μ$, either there exists a $ν$-almost surely invariant $μ$-measurable line field (corresponding do the stable distributions for almos…
Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity
本文引入Patterson-Sullivan系统，研究其在随机游走和熵刚性中的应用，证明了Tukia的测度边界刚性定理的推广，并探讨了Patterson-Sullivan测度的奇异性猜想。
This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teresópolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for aff…
The notion of slow entropy, both upper and lower slow entropy, was defined by Katok and Thouvenot as a more refined measure of complexity for dynamical systems, than the classical Kolmogorov-Sinai entropy. For any subexponential rate function $a_n(t)$, we prove there exists a generic class of invertible measure preserving systems such that the lower slow entropy is zero and the upper slow entropy is infinite. Also, given any subexponential rate $a_n(t)$, we show there exists a rigid, weak mixing, invertible system such that the lower slow entropy is infinite with respect to $a_n(t)$. This give…
Given a convex representation $ρ:Γ\to\textrm{PGL}(d,\mathbb{R})$ of a convex co-compact group $Γ$ of $\mathbb{H}^k$ we find upper bounds for the quantity $αh_ρ,$ where $h_ρ$ is the entropy of $ρ$ and $α$ is the Hölder exponent of the equivariant map $\partialΓ\to\mathbb{P}(\mathbb{R}^d).$ We also give rigidity statements when the upper bound is attained. We then study Hitchin representations and prove that if $ρ:π_1Σ\to\textrm{PSL}(d,\mathbb{R})$ is in the Hitchin component then $αh_ρ\leq 2/(d-1)$ (where $α$ is the Hölder exponent of the map $ζ:\partial\mathbb{H}^2\to\mathscr{F}$) with equalit…
Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences …
Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations
本文通过沿幂单方向的定量回复方法，给出了高阶对角化作用在 $SL_2$ 形式算术商上正熵测度的分类及其 Adelic 版本，并应用于整数倍丢番图逼近问题。
Rudolph-Johnson without dynamics: endomorphic rigidity on the circle
本文证明仅凭熵条件即可使圆周上的测度在亚空隙乘子自同态下趋于 Lebesgue 测度，给出有限熵界并推广 Rudolph-Johnson 定理，同时揭示 Lyons 猜想的成立与失败条件。
Katok's entropy conjecture near real and complex hyperbolic metrics
本文证明实或复双曲度量在负曲率度量空间中的局部邻域内，拓扑熵与 Liouville 熵相等当且仅当两度量相似，从而部分解决 Katok 熵刚性猜想，并推出双曲秩与 C^2 Anosov 叶状结构的局部刚性。
Local rigidity of Julia sets
本文给出判据，说明两个有理函数Julia集之间的局部同胚若满足一定正则性条件，则来自代数对应，并指出该结果将用于曲线动力学André-Oort猜想的证明。
Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces
本文研究了负曲率3-流形中高能 $k$-曲面的渐近行为，证明了拟 Fuchsian $k$-曲面增长率的刚性上界、标记能量谱的控制与刚性定理，并得出标记面积谱与能量谱渐近当且仅当截面曲率为常数。
We prove $\times a$ $\times b$ measure rigidity for multiplicatively independent pairs when $a\in\mathbb{N}$ and $b>1$ is a ``specified'' real number (the $b$-expansion of $1$ has a tail or bounded runs of $0$'s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along $\times b$ orbits. We also prove a quantitative version of this decay under stronger conditions on the $\times a$ invariant measure. The quantitative version together with the $\times b$ invariance of the limit measure is a step toward a general Host-type po…
Furstenberg discretization with moment control and entropy rigidity
针对局部紧群的余紧格，本文构造了满足任意阶矩条件的Furstenberg离散化，关联了Lyapunov指数与谱半径，推导了相关熵公式，得到Poisson边界保持结果并刻画了平稳作用的Zimmer可和性。
Margulis Multiverse: Infinitesimal Rigidity, Pressure Form and Convexity
本文研究高维无尖点 Margulis 时空模空间，构造压力形式，借助带标记 Margulis 不变量谱的无穷小确定性证明其在恒定熵截面上诱导黎曼度量，并确立固定线性部分时这些截面的凸性。
Kummer Rigidity for Hyperkähler Automorphisms
本研究针对射影超Kähler流形上具有正拓扑熵且以体积测度为最大熵测度的全纯自同构，通过Jensen不等式证明分布的一致伸缩性及流形平坦性，结合已有结果推出其为Kummer型例子，部分推广了复曲面对应结论。
We introduce and characterize extremal 2-cuts for good fractal necklaces. Using the characterization and the related topological properties of extremal 2-cuts, we prove that every good necklace has a unique necklace IFS in a certain sense. Also, we prove that two good necklaces admit only rigid homeomorphisms and thus the group of self-homeomorphisms of a good necklace is countable. In addition, a certain weaker co-Hopfian property of good necklaces is also obtained.
We prove that a topological homeomorphism conjugating two generic 1-parameter unfoldings of 1-variable complex analytic resonant diffeomorphisms is holomorphic or anti-holomorphic by restriction to the unperturbed parameter. We provide examples that show that the genericity hypothesis is necessary. Moreover we characterize the possible behavior of conjugacies for the unperturbed parameter in the general case. In particular they are always real analytic outside of the origin. We describe the structure of the limits of orbits when we approach the unperturbed parameter. The proof of the rigidity …
Multifractal Analysis, Livšic Rigidity, and Fluctuation Theorems for Axiom A Diffeomorphisms: The Pesin Formula and the Gallavotti-Cohen Symmetry
本文研究Axiom A微分同胚的多标度分析、Livšic定理及波动定理，推导了Pesin熵公式和Gallavotti-Cohen对称性，并探讨了压力的Legendre变换与Hausdorff维数的关系。
Local unmarked length spectrum rigidity for hyperbolic surfaces
本文证明严格1/9-夹紧的闭负曲率曲面若与双曲度量共享无标记长度谱则必为双曲，并由此得出双曲度量在C^2邻域内由该谱唯一决定。
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