2015-06-29

Metric Diophantine approximation driven by dynamical systemsIn a measure theoretical dynamical system $(X, \mathcal{B}, T, \mu)$, Poincar\'{e}'s recurrence theorem states that for any measurable set $A$ with positive $\mu$-measure, for almost all points in $x\in A$, its orbit $\{T^n(x)\}_{n\ge 1}$ will return back to $A$ infinitely often. If there is a compatible metric $d$, one can conclude that almost surely,
$\liminf_{n\to \infty}d(T^n(x), x)=0.$
This is a qualitative result in nature. So, one is led to consider a more quantitative Diophantine-like question: how about the size of the set
$W(\phi):=\Big\{x\in X: d(T^nx, x)<\phi(n), \ {\text{infinitely often}}\ n\in \mathbb{N}\Big\}$
in the sense of measure and fractal dimensions. In this talk, I present some results about the Hausdorff dimension of $W(\phi)$ in some concrete systems: Gauss expansion, beta expansion, as well as conformal iterated function systems. But the question is still open as far as a general system is concerned.
This is a joint work with Bo Tan and Jun Wu.

Difference of Primes and Related problems

Let $C_k(S)$ be the set of all common differences of arithmetical progeression with lengh k+1 appearing in a big subset S of natural numbers. We will introduce some results about the structure of $C_K(S)$. And also we will review some results on nonconventinal ergodic average, difference set of primes (Pintz ect.) and our some observation.

Uniform sets for infinite measure-preserving systems

The concept of a uniform set is introduced for an ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space. The uniform sets exist inasmuch as they generate the underlying sigma-algebra. This leads to the result that every ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space is isomorphic to a minimal homeomorphism on a locally compact metric space which admits a unique, up to scaling, invariant Radon measure.

Symbolic extension of Cantor minimal systems and order embeddig of dimension groups

In this talk we show that for given Cantor miniamal system (Y, T) and dimension group G satisfying that the dimension group H arizing from (Y,T) is order embedded into G and (Y,T) and G have some properties, there exists a Cantor minimal system (X,S) such that (1) (X,S) is a sybolic extension of (Y,T) and the dimension
group of (X,S) is order isomorphic to G. This problem is so-called `symbolic dynamical extension realization
of dimension groups'.

Birkhoff sum fluctuations of substitution dynamical systems

In this talk we will discuss deviation of Birkhoff sums for substitution dynamical systems with an incidence matrix having eigenvalues of modulus 1. Especially we will describe central limit theorem for fixed points of substitution. This is a joint work with E. Paquette.

Random sequence on Cantor carpet

I have constructed a low discrepancy sequence on any dimension using dynamical system.
Extending this results, I will construct a random sequence on the cantor carpet.

Arithmetic conditions on self-similar subsets

Let $E,F\subset\mathbb R^d$ be two self-similar sets. Let $\{\rho_i\}$
and $\{\gamma_j\}$ be the contraction ratios of $E$ and $F$,
respectively. Suppose that $E$ satisfies the SSC and $F\subset E$. Under
certain circumstances, we prove that there exist non-negative rational
numbers $t_{i,j}$ such that $\gamma_j=\prod_i\rho_i^{t_{i,j}}$.

2015-06-30

Aperiodic shifts of finite type of the Heisenberg groupWe describe a construction of a strongly aperiodic shift of finite type for the Heisenberg group. The talk will introduce a general framework to approach such problems and some conjectures for how to extend the construction more generally. This is joint work with M. Schraudner (Center for Mathematics and Modeling, Chile) and I. Ugarcovici (DePaul).

The error term of The Prime Orbit Theorem for expanding semiflows

We consider suspension semiflows of angle multiplying maps on the circle and study the distributions of periods of their periodic orbits. Under generic conditions on the roof function, we give an asymptotic formula on the number $\pi(T)$ of prime periodic orbits with period $\le T$. The error term is bounded, at least, by
\[
\exp\left(\left(1-\frac{1}{4\lceil \chi_{\max}/h_{\top}\rceil}+\varepsilon\right) h_{\top}\cdot T\right)\qquad \mbox{in the limit $T\to \infty$}
\]
for arbitrarily small $\varepsilon>0$, where $h_{\top}$ and $\chi_{\max}$ are respectively the topological entropy and the maximal Lyapunov exponent of the semiflow.

Sensitivity revisited

In this talk I will define a stronger form of sensitivity with respect to a family. Characterizations will be given for several well known families, including IP, finite IP and thick families. This is a joint work with Tao YU.

Dynamics on virtual contact structures

We consider closed 3-manifolds that admit a virtual contactstructure and prove existence of periodic Reeb orbits in the presence ofovertwisted disks. In order to get examples we define the Lutz twist forvirtual contact structures.

Weighted SMB theorem and weighted entropy for amenable group actions

Recently, Feng and Huang setup the theory of weighted entropy and pressure for finite many factor systems. In this talk, we consider weighted entropy for countable discrete amenable group actions. By giving suitable definitions of weighted partition and weighted Bowen ball, we prove a weighted version of SMB theorem and variational principle for entropy.

Sufficient conditions for the uniqueness of Sinai-Ruelle-Bowen measures

In this talk we present sufficient conditions for the uniqueness of Sinai-Ruelle-Bowen measures (SRB measures) as follows: (1) if a diffeomorphism is topologically conjugate to some transitive Anosov diffeomorphism, then there exists at most one SRB measure; (2) if a diffeomorphism of a $3$-dimensional manifold satisfies the specification property, then there exists at most one SRB measure.

Large deviations for systems with non-uniform structure

We use a weak Gibbs property and a weak form of specification to derive level-2 large deviations principles for symbolic systems equipped with a large class of reference measures. This has applications to a broad class of coded systems, including beta-shifts, S-gap shifts, and their factors.

Henon renormalization in three dimensional dynamics

Period doubling renormalization of strongly dissipative analytic Hénon-like map introduced by Lyubich and Martens has universality but non-rigidity. These properties are also true in certain invariant spaces under renormalization operator which appear only on three or higher dimensional dynamics. However, it is empathized that each invariant space is found by totally different techniques and conditions.

Distribution of Patches in Tilings and Corresponding Dynamical Systems

A tiling is a cover of R^d by tiles such as polygons that overlap only on their borders. A patch is a pattern consisting of finitely many tiles that appears in tilings. From a tiling, one can construct a dynamical system which encodes the nature of the tiling. I will talk about a relation between distribution of patches in tilings and properties of the corresponding dynamical systems.

2015-07-01

Entropy estimation In a symbolic dynamical system which is ergodic, the entropy is estimable from a single path and many estimators as this have been proposed so far. Here, we propose another one which is not only a consistent estimator of the entropy but also represent the complexity of finite sample paths well. It can be used as a criterion of randomness of finite words over an alphabet.

Beta expansions involving rotational action

Beta expansion gives a bridge between number theory and ergodic theory. Its absolutely continuous invariant measure (ACIM) was made explicit by Parry and it is equivalent to the 1-dim Lebesgue measure. We discuss a generalization of beta expansion involving rotation acting on the complex plane. We can show that the ACIM is unique if the expansion constant is larger than some explicit constant. We also discuss its associated symbolic dynamics. This is a joint work with Jonathan Caalim.

On the dual map of the Rauzy induction

We introduce a special type of interval exchange which is called an IPR map and define an induction on the set of IPR maps. This is a simple modification of an induction introduced by Cruz and da Rocha. We show that our induction gives the dual of the sense of Schweiger.

On the number of fixed points of $D_{\infty}$-Markov chains

The number of fixed points, the Lind zeta functions, $D_{\infty}$-strong shift equivalence ($D_{\infty}$-SSE), and $D_{\infty}$-shift equivalence ($D_{\infty}$-SE) are conjugacy invariants for $D_{\infty}$-Markov chains. The relationships between them are illustrated: Two $D_{\infty}$-Markov chains are conjugate if and only if there is a $D_{\infty}$-SSE (lag $2k$) between their flip pairs for some positive integer $k$. However, the existence of a $D_{\infty}$-SE between two flip pairs does not imply that the corresponding $D_{\infty}$-Markov chains have the same Lind zeta functions. Similarly, if two $D_{\infty}$-Markov chains have the same Lind zeta functions, it does not imply the existence of a $D_{\infty}$-SE between their flip pairs. However, if the defining matrices are irreducible with real eigenvalues, such that their characteristic polynomials are products of an irreducible polynomial in $\mathbb{Z}[t]$ and some powers of $t$ and the numbers of fixed points are all nonzero, then if two $D_{\infty}$-Markov chains share the same Lind zeta functions, it guarantees the existence of a $D_{\infty}$-SE between their flip pairs. Moreover the numbers of fixed points determine certain partial entry sums of the matrices which are related to the corresponding $D_{\infty}$-SE.

An SDE approach to leafwise diffusions on foliated spaces

We can regard foliated spaces as generalization of dynamical systems.
Leafwise diffusions and harmonic measures play important roles in the ergodic theory
of foliated spaces. In this talk, we introduce stochastic differential equations on a
compact foliated space to construct leafwise diffsuions. This construction of
leafwise diffusions provides good approaches to some problems (the stochastic continuity
with respect to starting points, a central limit problem for these processes and so on).

Weak measure expansive flows

Recently a notion of measure expansivity for flows was introduced by Carrasco and Morales in [1] as a generalization of expansivity, and they proved that there were no measure expansive flows on closed surfaces. In this talk we introduce a concept of weak measure expansivity for flows which is really weaker than that of measure expansivity, and show that there are weak measure expansive flows on closed surfaces. Moreover we prove that any $C^1$ stably weak measure expansive flows on a $C^{\infty}$ closed manifold $M$ satisfies both Axiom $A$ and no-cycle condition; and any $C^1$ stably measure expansive flows on $M$ is quasi-Anosov.
(*) Joint work with K. Lee.

Criteria of measure-preservation of 1-Lipschitz functions on \Fq[[T]] in terms of four bases

We gave a criteria of the measure-preservation of 1-Lipshcitz functions on $\Fq[[T]]$ in terms of the van der Put expansion and use this criteria to characterize the measure-preservation of 1-Lipshcitz functions on $\Fq[[T]]$ in terms of the three well-known bases: Carlitz polynomials, digit derivatives, and digit shifts.
This is joint work with Youngho Jang and Chunlan Li.

Artin-Mazur zeta functions and lap counting functions of generalized $\beta$-transformations

We consider the generalized $\beta$-transformation, introduced by Pawl G\"ora in 2007 and study the analytic properties of its Artin-Mazur zeta function and lap counting function.
We show that both functions are holomorphic in the open disk whose radius is $1/\beta$ and have a simple pole at $1/\beta$. We further show that each of them can be extended to a meromorphic function which is expressed by using coefficients of $f$-expansion of 1.

On sofic-like shifts and flips for a synchronized system

In this talk, we will be interested in coded systems and almost sofic shifts. These shift spaces generalize sofic shifts. An irreducible sofic shift is both coded and almost sofic.
We investigate which properties of irreducible sofic shifts are extended to coded systems and almost sofic shifts.
We will also be interested in the following property (P): (P) if a shift space has a flip, then it has infinitely many non-conjugate ones. We prove that if an infinite synchronized system has a flip, then there are countably many non-conjugate ones. Every synchronized system is coded. However, the property (P) can not be extended to
coded systems.

2015-07-02

Multifractal Analysis for the Complex Analogues of the Devil's Staircase and the Takagi Function in Random Complex DynamicsWe consider random dynamical systems of rational maps on the Riemann sphere. We investigate the function $T$ of probability of tending to one minimal set and its partial derivative $C$ with respect to the probability parameter. It turns out that under certain conditions, the function $T$ is a complex analogue of the devil's staircase or Lebesgue's singular functions and the function $C$ is a complex analogue of the Takagi function. Those functions $T$ and $C$ are continuous on the Riemann sphere and vary precisely on the Julia set of the associated semigroup, which is a thin fractal set. We investigate the multifractal analysis for the pointwise Holder exponents of these functions. This is a joint work with Johannes Jaerisch (Waseda University).

On aperiodic hexagonal tilings

There has been a long search for a single prototile which tiles the plane only in aperiodic ways, This tile is called aperiodic mono-tile.
A few years ago, Taylor and Socolar had introduced an aperiodic mono-tile. We call the tile Taylor-Socolar tile. On the other hand, much earlier, Penrose had introduced a mono-tile with some matching rule which tile the plane only aperiodically. Due to the matching rule, his tile had not got much attention before Taylor-Socolar tile was introduced. Although they build different tilings, the two tiles are based on hexagonal tiles and have a lot of common features,
In this talk, we consider tilings built from these tiles and study dynamical systems arising from them. We will discuss not only difference but also similarity and close connection between these two tilings.

Measures with maximum total exponent of C1 diffeomorphisms with basic sets

A measure with maximum total exponent is an invariant Borel probability measure maximizing the integral of a certain function depending on a dynamical system. We state that any $C^1$ diffeomorphism with a basic set has a $C^1$ neighborhood satisfying the following properties. A generic element in the neighborhood has a unique measure with maximum total exponent which is of zero entropy and fully supported on the continuation of the basic set. To the contrary, we state that if $r \geq 2$ then any $C^r$ diffeomorphism with a basic set does not have a $C^r$ neighborhood satisfying the above properties.

Homoclinic classes of C1-vector fields

Let $\gamma$ be a hyperbolic closed orbit of a $C^1$ vector field $X$ on a compact boundaryless Riemannian manifold $M$, and let $H_X(\gamma)$ be the homoclinic class of $X$ containing $\gamma$. In this talk, we show that if the
homoclinic class $H_X(\gamma)$ is $C^1$-persistently expansive and shadowable, then it is hyperbolic. Moreover, we prove that $C^1$-generically, $H_X(\gamma)$ is hyperbolic if it is shadowable and locally maximal.
(*) Joint work with M. Lee.

Exponential Mixing and Effective Counting in Trees

We consider the space of all bi-infinite geodesics in a graph of groups, the geodesic translation map and the equilibrium measure on the space with a given potential. We introduce the Markov chain associated with the dynamical system of geodesic translation map and prove that it is exponentially mixing if the fundamental group of the graph of groups has weighted spectral gap property. We can apply this result to count effectively the number of closed geodesics with weights given by the potential.

Higher dimensional Frobenius problem: maximal saturated cone, growth function and rigidity

We consider $m$ integral vectors $X_1, \cdots, X_m \in \mathbb{Z}^s$ located in a half-space of $\mathbb{R}^s$ $(m \ge s \ge 1)$ and study the structure of the additive semi-group $X_1\mathbb{N} +\cdots+ X_m\mathbb{N}.$ We introduce and study maximal saturated cone and directional growth function which describe some aspects of the structure of the semi-group. When the vectors $X_1,\cdots, X_m$ are located in a fixed hyperplane, we obtain an explicit formula for the directional growth function and we show that this function completely characterizes the defining data $(X_1,\cdots,X_m)$ of the semi-group. The last result will be applied to the study of Lipschitz equivalence of Cantor sets(see [Rao and Zhang, Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets, Preprint 2015]).