Could you teach me an example of NOT uniquely ergodic but ergodic transformation? And when any continuous, measurable, and ergodic transformation on a topological space X is uniquely ergodic, how topological properties X has?
2026-03-27 02:33:18.1774578798
Not uniquely ergodic transformation
606 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in DYNAMICAL-SYSTEMS
- Stability of system of parameters $\kappa, \lambda$ when there is a zero eigenvalue
- Stability of stationary point $O(0,0)$ when eigenvalues are zero
- Determine $ \ a_{\max} \ $ and $ \ a_{\min} \ $ so that the above difference equation is well-defined.
- Question on designing a state observer for discrete time system
- How to analyze a dynamical system when $t\to\infty?$
- The system $x' = h(y), \space y' = ay + g(x)$ has no periodic solutions
- Existence of unique limit cycle for $r'=r(μ-r^2), \space θ' = ρ(r^2)$
- Including a time delay term for a differential equation
- Doubts in proof of topologically transitive + dense periodic points = Devaney Chaotic
- Condition for symmetric part of $A$ for $\|x(t)\|$ monotonically decreasing ($\dot{x} = Ax(t)$)
Related Questions in ERGODIC-THEORY
- the mathematics of stirring
- Kac Lemma for Ergodic Stationary Process
- Ergodicity of a skew product
- Is every dynamical system approaching independence isomorphic to a Bernoulli system?
- Infinite dimensional analysis
- Poincaré's Recurrence Theorem
- Chain recurrent set is the set of fixed points?
- Is this chaotic map known?
- A complex root of unity and "dense" property of the its orbit on the unit circle
- Books on ergodic operators
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
For a transformation to have a unique ergodic measure is not that common as you seem to think. In fact for many examples (I would say for the vast majority) many ergodic measures exist. This is due to the fact the often there are periodic orbits giving rise to ergodic measures. Here are some classes of examples:
Rational rotations of the circle: There every (finite) orbit can be used as the support of an ergodic measure (the uniform distribution on this orbit).
Full shifts: Again there are many periodic orbits, so put the uniform measure on one of them and you get an ergodic measure. However there are many other ergodic measures for such a system (Bernoulli measures, Markov measures, Gibbs measures etc.).
Similarly endomorphisms of the circle, hyperbolic toral automorphisms, Markov shifts, sofic shifts etc. etc. etc. all have multiple ergodic measures.
On the contrary, the standard classes of uniquely ergodic systems cyclic permutations on a finite space, irrational rotations on the circle (a uncountable connected space) or more generally on a compact (abelian) topological group, sturmian subshifts or subshifts coming from primitive substitutions (both have totally disconnected phase spaces) etc.
So taking those uniquely ergodic examples, your second questions does not seem to have a good answer with respect to topology. Unique ergodicity has to do less with the topological properties of the phase space and more with the "rigidity" of the transformation.
There is one condition however: Think about what it would mean for a space to only allow for uniquely ergodic transformations. As I explained above more than one finite orbit is bad. So what are the spaces where no continuous transformation can have multiple finite orbits? Think of the most basic transformation, the one definable on every phase space...