**Dynamical formulation of Prisoner’s dilemma**

Originally, consider the two players, each has a set of stratagies, say and . The pay-off for player depends on the choices of both players.

Now consider two dynamical systems . The set of stratagies consists of the invariant probability measures, and the pay-off functions can be

— , where ;

— .

The frist one is related to Ergodic optimization. The second one does sound better, since one may want to avoid a complicated (measured by its entropy) stratagy that has the same pay-off.

**Gambler’s Ruin Problem**

A gambler starts with an initial fortune of $i,

and then either wins $1 (with ) or loses $1 (with ) on each successive gamble (independent of the past). Let denote the total fortune after the n-th gamble. Given , the gambler stops either when (broke), or (win), whichever happens first.

Let be the stopping time and be the probability that the gambler wins. It is easy to see that and . We need to figure out for all .

Let , and . There are two cases according to :

— (prob ): win eventually with prob ;

— (prob ): win eventually with prob .

So , or equivalently,

(since ), .

Recall that and . Therefore , . Summing over , we get , (if ) and (if ). Generally (if ) and (if ).

Observe that for fixed , the limit only when , and whenever .

**Finite Blaschke products**

Let be an analytic function on the unit disc with a continuous extension to with . Then is of the form

,

where , and is the multiplicity of the zero of . Such is called a finite Blaschke product.

**Proposition. ** Let be a finite Blaschke product. Then the restriction is measure-preserving if and only if . That is, for some .

Proof. Let be an analytic function on . Then and .

Significance: there are a lot of measure-preserving covering maps on .

**Kalikov’s Random Walk Random Scenery**

Let , and to the shift . More generally, let be a finite alphabet and be probability vector on , and , . Consider the skew-product , . It is clear that preserves any , where is -invariant.

**Proposition. ** Let . Then for all .

Proof. Note that . CLT tells that as , where as . There are only different -strings (up to an error).

Significance: this gives a natural family of examples that are K, but not isomorphic to Bernoulli.

** Creation of one sink. **1D case. Consider the family , where . Let the first parameter such that the graph is tangent to the diagonal at . Note that is parabolic. Then for , has two solutions , where is a sink, and is a source.

2D case. Let be a rectangle, be a diffeomorphism such that is a horseshoe lying above of shape ‘V’. Moreover we assume . Let such that is the regular horseshoe intersection: V . Clearly there exists a fixed point of in . We assume . Then Robinson proved that admits a fixed point in which is a sink.

First note that for any , and any fixed point of (if exists), it is not on the boundary of . Since is a nondegenerate fixed point of , the fixed point continues to exist for some open interval (assume it is maximal, and denote the fixed point by ). Clearly . Note that is also fixed by , since it is a closed property. If there is some moment with for the fixed point of , then it is already a sink, since . So in the following we consider the case for all , . Then the continuous dependence of parameters implies that both are continuous functions of . The fixed point must be degenerate, since the fixed point ceases to exist beyond , which means: for some .

Case 1. . Note that . So for some , which implies that for some . In particular, , and . So is a (complex) sink.

Case 2. . Note that . Similarly for some .

So in the orientation-preserving case there always exists a complex sink. In the orientation-reversing case (), we need modify the argument for case 2:

Case 2′. . Note that . So for some . We pick close to in the sense that , which implies , too. So is also a sink.