Expansive measures for flows

Continuum-wise expansive measures

This thesis is in two parts. The first part consists of chapters 1-4 and the second part is chapter 5. In the first part we consider the idea of approximating pieces of orbits by a single orbit. There are many examples of such properties (approximation property (A.P.) see chapter one, definitions 1.2, 1.4), (Specification property (S.P.),[24]), (pseudo orbit tracing property (P.O.T.P.) [25] and definition 4.1). In chapter one, we show that (A.P.) for a homeomorphism (flow) is equivalent to topological transitivity and density of periodic points and how this property (A.P.) is invariant under topological conjugacy. In theorem 1 we prove that an expansive homeomorphism which is topologically mixing and has P.O.T.P. also has the S.P. In chapter two, theorem 2, we prove that the P.O.T.P. for flows is invariant under topological conjugacy with preserved orientation (velocity changes). Also we prove in theorem 3 that the suspension flow [2] for a homeomorphism T:X → X has P.O.T.P. if and only if T has the P.O.T.P. In chapter three, we prove that an expansive flow which has the P.O.T.P. is topologically stable (Theorem 4). In chapter four, the last theorem in this part is that every flow without fixed points on a compact manifold M which is topologically stable has the P.O.T.P. Then some important corollaries are deduced. In the second part of this thesis … [please see full thesis for rest of abstract]

In this paper, we study the centralizer of a separating continuous flow without fixed points. We show that if M is a compact metric space and ϕt:M→M is a separating flow without fixed points, then ϕt has a quasi-trivial centralizer, that is, if a continuous flow ψt commutes with ϕt, then there exists a continuous function A:M→R which is invariant along the orbit of ϕt such that ψt(x)=ϕA(x)t(x) holds for all x∈M. We also show that if M is a compact Riemannian manifold without boundary and Φu is a homogenous separating C1 Rm-action on M, then Φu has a quasi-trivial centralizer, that is, if Ψu is a Rm-action on M commuting with Φu, then there is a continuous map A:M→Mm×m(R) which is invariant along orbit of Φu such that Ψu(x)=ΦA(x)u(x) for all x∈M. These improve Theorem 1 of [M. Oka, Expansive flows and their centralizers, Nagoya Math. J. 64 (1976), pp. 1–15.] and Theorem 2 of [W. Bonomo, J. Rocha, and P. Varandas, The centralizer of Komuro-expansive flows and expansive Rd-actions, Math. Z. 289(3–4) (2018), pp. 1059–1088.] respectively.

In this paper, we consider positively weak measure expansive homeomorphisms and flows with the shadowing property on a compact metric space X. Moreover, we prove that if a homeomorphism (or flow) has a positively weak expansive measure and the shadowing property on its nonwandering set, then its topological entropy is positive.

Let $\alpha_0$ be an affine action of a discrete group $\Gamma$ on a compact homogeneous space $X$ and $\alpha_1$ a smooth action of $\Gamma$ on $X$ which is $C^1$-close to $\alpha_0$. We show that under some conditions, every topological conjugacy between $\alpha_0$ and $\alpha_1$ is smooth. In particular, our results apply to Zariski-dense subgroups of $SL_d(\mathbb{Z})$ acting on the torus $\mathbb{T}^d$ and Zariski-dense subgroups of a simple noncompact Lie group $G$ acting on a compact homogeneous space $X$ of $G$ with an invariant measure.

OF THE DISSERTATION Some parabolic and elliptic problems in complex Riemannian geometry by Bin Guo Dissertation Director: Professor Jian Song This dissertation consists of three parts, the first one is on the blow-up behavior of Kahler Ricci flow on CP blown-up at one point, and the second one on the convergence of Kahler Ricci flow on minimal projective manifolds of general type, and the last one is on the existence of canonical conical Kahler metrics on toric manifolds. In the first part, we consider the Ricci flow on CP blown-up at one point starting with any rotationally symmetric Kahler metric. We show that if the total volume does not go to zero at the singular time, then any parabolic blow-up limit of the Ricci flow along the exceptional divisor is a non-compact complete shrinking Kahler Ricci soliton with rotational symmetry on Cn blown-up at one point, hence the FIK soliton constructed in [36]. In the second part, we consider the Kahler Ricci flow on a smooth minimal model of general type, following the ideas of Song ([76, 77]), we show that if the Ricci curvature is uniformly bounded below along the Kahler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci curvature lower bound assumption, the Gromov-Hausdorff limit of the flow is homeomorphic to the canonical model of the manifold. Moreover, we will give a purely analytic proof of a recent result of Tosatti-Zhang ([102]) that if the canonical line bundle KX is big and nef, but not

In this paper, we give an equivariant compactification of the space $${\mathbb {P}}{\text {Flat}}(\Sigma )$$ of homothety classes of half-translation structures on a compact, connected, orientable surface $$\Sigma $$ . We introduce the space $${\mathbb {P}}{\text {Mix}}(\Sigma )$$ of homothety classes of mixed structures on $$\Sigma $$ , that are $${\text {CAT}}(0)$$ tree-graded spaces in the sense of Drutu and Sapir, with pieces which are $${\mathbb {R}}$$ -trees and completions of surfaces endowed with half-translation structures. Endowing $${\text {Mix}}(\Sigma )$$ with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that $${\mathbb {P}}{\text {Mix}}(\Sigma )$$ is an equivariant compactification of $${\mathbb {P}}{\text {Flat}}(\Sigma )$$ , thus allowing us to understand in a geometric way the degenerations of half-translation structures on $$\Sigma $$ . We finally compare our compactification to the one of Duchin–Leininger–Rafi, based on geodesic currents on $$\Sigma $$ , by the mean of the translation distances of the elements of the covering group of $$\Sigma $$ .

For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

formula omitted]-continuity, impatience and myopia for Choquet multi-period utilities

Unique equilibrium states for flows and homeomorphisms with non-uniform structure

Rapid onset vasodilation (ROV) following single muscle contractions has been examined in the forearm of humans, but has not been characterized in the leg. Given known vascular differences between arms and legs as well as the greater susceptibility of legs to atherosclerosis, we sought to characterize ROV in the leg. To do this, 9 healthy men performed random ordered single muscle contractions at 5, 10, 20, 40, and 60% of their maximum voluntary contraction (MVC) in the leg (knee extension) and forearm (handgrip). Continuous blood flow (BF) measures were made via duplex Doppler‐ultrasound from the brachial and femoral artery during 15 sec prior to and 45 sec after release of each contraction. Three trials were performed at each intensity. Limb BF was normalized to lean mass of forearm and thigh region through DEXA. Single muscle contractions produced an intensity dependent increase in BF, peaking ~4 sec post contraction in both limbs. Initial BF responses (1st heart beat) were significantly greater in the leg, most notable at lower MVC intensities (e.g. 10%‐MVC initial response; Arm=+19±5% vs. Leg=+75±15%, P=0.01). However, peak changes in BF were similar between limbs for each contraction intensity (e.g. 60%‐MVC peak; Arm=+207±33% vs. Leg=+208±33%, P>0.05). These preliminary findings indicate that peak BF responses to single muscle contractions are similar between limbs establishing the presence of ROV in the leg.Grant Funding Source: NIH grant RO1 HL093167 (PJF)